











































































































































PLANE 

TRIGONOMETRY 


BY 

WILLIAM L!' HART, Ph.D. 


PROFESSOR OF MATHEMATICS 
UNIVERSITY OF MINNESOTA 




D. C. HEATH AND COMPANY 


BOSTON NEW YOKE CHICAGO 

ATLANTA SAN FRANCISCO DALLAS 

LONDON 


C. 







Copyright, 1933 
By W. L. Hart 


No part of the material covered by this 
copyright may be reproduced in any form 
without written permission of the publisher. 

3 e 3 


PRINTED IN THE UNITED STATES OF AMERICA 


JUN 12 1933 


©CIA 


62943 



PREFACE 


This book offers a concise treatment of plane trigonometry which 
gives full recognition to both the numerical and the analytical sides of 
the subject, and yet allows the instructor the maximum latitude in his 
teaching. The flexible nature of the text suits it for courses of various 
lengths and different objectives without introducing any discontinuity 
as judged from the standpoint of one who studies the complete book. 
The chapters on the solution of triangles emphasize modern points of 
view with respect to numerical computation and include a wealth of 
practical applications. On the other hand, the sections devoted to 
analytical trigonometry are definitely oriented with respect to their ap¬ 
plications in more advanced mathematics. The tables provided with 
the book are particularly complete and convenient. A comfortably 
large page was employed in order to permit the incorporation of several 
useful devices in the tables and to insure maximum legibility on their 
most important pages. Special attention is directed to the following 
features of the text and tables. 

The trigonometry of the acute angle and applications are presented 
before the general angle is introduced. However, if desired, part of the 
introductory chapter on the general angle may be studied without lack 
of continuity before the chapters on the acute angle are considered. 

The natural values of secants and cosecants are included in both the 
four-place and the five-place tables. This is done in recognition of the 
usefulness of these functions in non-logarithmic computation, which is 
becoming increasingly important due to the widespread use of computing 
machines by practicing engineers. 

The four-place tables include all necessary columns of proportional 
parts so that interpolation in these tables may be performed as conven¬ 
iently as in the usual five-place tables. This permits the instructor to 
select either four-place or five-place computation for major emphasis, 
with the assurance that the student can carry his acquired skill over to 
the other variety of computation. In appropriate places, separate prob¬ 
lems, or separate answers, are provided for both four-place and five-place 
computation. 

In the five-place tables of logarithms of the functions, proportional 
parts are provided for interpolation to tenths of a minute. 

The problem material is extremely extensive and carefully graded. 
In each of the key exercises, the problems are so numerous that, under 

iii 


iv PREFACE 

normal conditions, three different satisfactory assignments could be made 
from the exercise. 

Problems particularly designed for use during the class hour are pro¬ 
vided at the beginning of practically every exercise. 

Review exercises appear at strategic points throughout the text. A 
final summarizing review of 200 problems is given at the end of the book. 

Identities and equations receive the heavy emphasis which they de¬ 
serve, but the chapters on these topics are arranged to facilitate various 
stages of economy in the time devoted to this part of the course. Each 
of these chapters is thoroughly graded, with a long miscellaneous exercise 
at the end to provide material for even the most intensive course. 

Topics which present special difficulty, or which are not essential in a 
minimum course, are clearly marked supplementary, or are indicated by a 
star, ★. The omission of such material does not disturb the continuity 
of the remainder of the text. 

A complete chapter on logarithms is included. 

Answers are given in the text only for odd-numbered problems so as 

to offer the instructor the greatest latitude. A separate answer book for 
the even-numbered problems is provided free-of-charge for each student 
when requested by the instructor. Either the odd-numbered problems 
or the even-numbered ones form a satisfactory basis for the course. 

Illustrative examples are used extensively to illustrate new theory, 
to recall previous knowledge, and to furnish models for the student’s own 
solutions. 

Frequent and clearly displayed summaries are introduced to aid the 
student in future references and to systematize his methods. 

Carefully arranged pages automatically divide the text into easily 
recognized lesson units. 

In the tables, it will be found that Table X6 and the part of Table XI 
devoted to secants and cosecants present considerable novelty as com¬ 
pared with current sets of tables for student use. Each of the tables in 
this book was checked against highly accurate tables * with entries given 
to at least three more significant digits than were desired in this book. 

University op Minnesota 
April 15, 1933 

* The principal tables employed in checking were as follows: Logarithmic-Trigo¬ 
nometrical Tables with Eight Decimal Places, by Bauschinger and Peters; Sieben- und 
Mehrstellige Tafeln der Kreis- und Hyperbelfunctionen, by Keiichi Hayashi; Nouvelles 
Tables Trigonometric Fundamentales, by H. Andoyer. 


CONTENTS 


CHAPTER PAGE 

I. FUNCTIONS OF ACUTE ANGLES. 1 

II. LOGARITHMS. 20 

III. LOGARITHMIC SOLUTION OF RIGHT TRIANGLES . . 42 

IV. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE .... 56 

V. RADIAN MEASURE. 73 

VI. VARIATION AND GRAPHS OF THE FUNCTIONS ... 78 

VII. THE FUNDAMENTAL IDENTITIES. 89 

VIII. ADDITION FORMULAS AND RELATED TOPICS .... 101 

IX. OBLIQUE TRIANGLES.120 

X. POLAR COORDINATES.149 

XI. INVERSE TRIGONOMETRIC FUNCTIONS.152 

XII. COMPLEX NUMBERS AND DE MOIVRE’S THEOREM . 160 

GENERAL REVIEW.169 

APPENDIX . . . ..175 

INDEX.185 


v 
























PLANE TRIGONOMETRY 


CHAPTER I 

FUNCTIONS OF ACUTE ANGLES 

1. Introduction. In the history of mathematics we find that the 
field called trigonometry was first studied intensively because of its 
usefulness in surveying and astronomy. 

Illustration 1. Suppose that a man on the lower side of the river in Figure 1 
desires to find the distance AB without crossing the river. Then, he may proceed 
as follows. 

I. By use of surveying instruments, he 
locates a point C so that BC is perpen¬ 
dicular to BA. Then, he measures BC: 
suppose that BC = 100 feet. 

II. He moves to C and measures the 
angle M. Suppose that M = 50°. 

III. From a trigonometric table (to be 
met later), he finds that, because M = 50°, 

A B 

^ = 1.19; AB = 1.19(5(7) = H9 feet. 

B L 

The use of trigonometry in the indirect measurement of angles or 
distances, as in Illustration 1, furnishes ample justification for the 
study of the subject. However, this numerical side of it is by no 
means its only important feature today. The theoretical aspects 
of trigonometry are indispensable not only in advanced pure mathe¬ 
matics but also in many fields of applied mathematics. 

2. Useful terminology. In a given problem, a constant is a num¬ 
ber symbol whose value does not change during the discussion. A 
variable is a number symbol which may take on various values. 

Illustration 1. The area of a circle is given by the formula A = 7rr 2 , where r 
is the radius. In considering different circles, it is a constant, approximately 
3.1416 for all circles, but r and A are variables. 

It is customary to say that one variable, y, is a function of another 
variable, x, in case whenever a value is assigned to x a corresponding 
value of y can be determined. 



1 






2 


TRIGONOMETRY 


Illustration 2. On account of the relation A = irr 2 , we say that the area A of 
a circle is a function of the radius r because whenever a value is assigned to r we 
can determine a corresponding value of A. 


3. A property of right triangles. Let * a represent angle CAB in 
Figure 2. Right triangles ABC and ADF are similar and hence 

AB _ AD 
AC AF' 

Thus, the difference in size between triangles 
ABC and ADF has no effect on the ratio of the 
hypotenuse to the base; this ratio has the same 
value for all right triangles with a as the base 
angle. Similarly, we can state that the ratio of any two sides of a 
right triangle, with a as one acute angle, depends only on the size 
of a and not on the size of the triangle. 



Fig. 2 


4. Trigonometric functions. In Figure 3 let a represent any acutef 
angle. From any point B on either side of a drop a perpendicular 
BC to the other side, thus forming a right tri¬ 
angle ABC. Let the small letters a, b, and c 
represent the lengths of the sides respectively 
opposite to the vertices A, B, and C. Then, from 
Section 3 it follows that the values of the six ratios 
which are named below depend only on the size of a 
and not on the particular point B which was used. Fig. 3 


| is called the tangent of a. 

Abbreviation: 

tan a. 

^ is called the cotangent of a. 

Abbreviation: 

cot a. 

- is called the sine of a. 
c 

Abbreviation: 

sin a. 

- is called the cosine of a. 
c 

Abbreviation: 

cos a. 

- is called the secant of a. 

Abbreviation: 

sec a. 

- is called the cosecant of a. 
a J 

Abbreviation: 

esc a. 


* Usually we shall employ Greek letters to represent angles. The letters a, f3, y, d, 
and <fi are called alpha, beta, gamma, theta, and phi, respectively. 

f An acute angle is one between 0° and 90°. Functions of 0°, 90°, and of all other 
angles which are not acute will be considered in later chapters. 








FUNCTIONS OF ACUTE ANGLES 


3 


In Figure 3 we notice that a is the leg opposite to a, b is the leg 
adjacent to a, and c is the hypotenuse. Hence, if a is any acute angle, 
which is located as one angle in a right triangle, then 


a _ opposite side 
c hypotenuse ’ 
b _ adjacent side 
c hypotenuse ’ 


tan a = 


a _ opposite side 
b adjacent side’ 


esc a = 


sec a = 


cot a = 


c 

a 

c 

b 

b 

a 


hypotenuse 
opposite side’ 
hypotenuse 
adjacent side’ 
adjacent side 
opposite side 


(1) 


Illustration 1. 
in Figure 4, then 
sin a = f; 
esc a = f; 
sin |S = i; 
esc /3 = f; 


If a and /3 are the angles 


cos a = f; 
sec a = f; 
cos j8 = f; 
sec /3 = f ; 


tan a — f; 
cot a = i; 
tan j8 = i ; 
cot /3 = f. 



Fig. 4 


If a is any given angle, a corresponding set of values can be de¬ 
termined for the six ratios in (1). Hence, sin a, cos a, tan a, esc a, 
sec a, and cot a are functions of a. More explicitly, they are called 
trigonometric functions of a. Hereafter in this book, unless other¬ 
wise specified, when referring to a function of an angle we shall mean 
one of the six trigonometric functions defined in (1). 

Note 1. Recall the Pythagorean theorem: in a right triangle, the square of the 
hypotenuse equals the sum of the squares of the other two sides. In Figure 3, 

c 2 = a 2 + (2) 

Example 1. Find the functions of a in Figure 5. 

Solution. 1. We are given b = 5 and c = 6. From (2), 

6 2 = o 2 + 5 2 ; a 2 = 36 - 25 = 11. 

2. Hence, a = Vll. From equations 1, 

VTl 5 . Vll 6 

sin a = cos a = tan a = —g—; sec a = 

6 eVH eVn * , 5 5vTi 

V 11 V 11 V 11 11 Vll 11 



Comment. In Example 1, each function could be^ expressed as a decimal, 
to any desired number of places. Thus, since Vll = 3.317 (from Table I), 
sin a = |(3.317) = .553. 

The following example illustrates the fact that if we are given the 
value of one function of an unknown acute angle a, then we can con¬ 
struct the angle and obtain all of its functions. 

* The preceding algebraic operation rationalized the denominator of 6/ VlT. 














4 


TRIGONOMETRY 


Example 2. If sin a = f, find the other functions of a. 


Solution. 1. Refer to equations 1. Since sin a = ^ = -> we use a triangle 
with o = 3 and c — 4. 

2. From a 2 + b 2 = c 2 , b 2 = c 2 - a 2 ; 

b 2 = 16 - 9 = 7; b = V7. 

4 

3. From Figure 6 we read esc a = 

V7 3 3V7 

T’ 

V7 ^ ^ ___ 


cos a = —7-; tan a = 


cot a = -5-- sec a = — 

3 » V 7 


7 > 
4 4V7 



Comment. Instead of using o = 3 and c = 4, we could have used o = 6 and 
c = 8, or any other values such that | = To construct Figure 6, we could 

proceed as follows: draw an angle of 90°; from C, its vertex, lay off CB = 3; with 
B as center and radius equal to 4 units, strike an arc cutting CA at A] then angle CAB 
is the desired angle a. 


5. Useful properties of the trigonometric functions. In Figure 3 
we notice that a < c and b < c. Therefore, if a is any angle between 
0° and 90°, sin a and cos a are less than 1 because a/c and b/c are 
fractions whose denominator is greater than either numerator. Simi¬ 
larly, it is seen that sec a and esc a are greater than 1 , because the 
numerator in c/a and c/b is greater than either denominator. 


Note 1. Recall that the reciprocal of a number N is defined to be 1 /N. 
Thus, the reciprocal of 4 is I; of f is (1 -f- f), or f. The reciprocal of any fraction 


k 


1 -*• jr), or that is, to obtain the reciprocal of a fraction we invert the fraction. 


On inspecting equations 1 on page 3 we verify the following recip¬ 
rocal relations: 


cot a = 

1 

or 

tan a = 

1 

tan a 

cot a 

sin a = 

1 

or 

esc a = 

1 

esc a 

sin a 

cos a = 

1 

or 

sec a = 

1 

sec a 

cos a 


( 1 ) 


Illustration 1. In equations 1, page 3, cot a = ^ and tan a = hence 


cot a = 


- — In Example 2 above, sin a = § while esc a — 
tan a . * 4 3 












FUNCTIONS OF ACUTE ANGLES 


5 


EXERCISE 1 

Read the functions of a and of (3 from the right triangle: 



5 17 v 2 


Find the reciprocal of each quantity: 

9. 3. 10. 6. 11. h 12. f. 13. $. 14. t¥t. 


By mere inspection, tell the value of the unknown function: 


15. cos a = find sec a. 

16. cot a — f; find tan a. 

17. esc 0 = £; find sin 0. 

18. sec 0 = f; find cos 0 . 

19. tan /3 = A; find cot /3. 

20 . sin 7 = find esc 7 . 


21 . sec 7 = 6 ; find cos 7 . 

22 . esc j 8 = 10; find sin /3. 

23. cot 0 = 4; find tan 0. 

24. sin a = A; find esc a. 

25. cos (f) = find sec $. 

26. tan <j) = 18; find cot </>. 


Each of the following problems refers to a right triangle ABC lettered as in 
Figure 3, page 2 . Find the unknown side by use of the Pythagorean theorem 
and obtain all functions of a and of with all denominators rationalized. 


27. b = 5; a = 12. 

28. a = 8 ; b = 6 . 

29. c — 13; a = 5. 

30. c = 17; b = 8 . 

31. a = 7; b = 24. 


32. c = 25; a = 24, 

33. a = 6 ; c = 10. 

34. a = 1; b = 1. 

35. c = 2; a = l. 

36. a = V$; 6 = 1 . 


37. c = 5; o = 2. 

38. c = 3; 6 = 2 

39. c = V2; 6 = 1 

40. 6 = k; a = k. 

41. o = k; c = 2k. 


Construct the acute angle a and find its other functions: 

42. tan a = f. 44. sin a = i. 46. cot a = ^r. 48. sec a = xf. 

43. cot a = 45. cos a = yf- 47. esc a = If. 49. cos a = ^. 

50. sin a = i. 52. esc a = 6 . 54. sec a = V 2 . 

51. sec a = 5. 53. tan a = 1. 55. esc a = ^3. 

56. In Figure 3, page 2, if a < 45° then a < b. Make a statement 

which proves that if a is between 0° and 45° then tan a < 1 and 

cot a > 1. Similarly, prove that if a is between 45° and 90° then 
tan a > 1 and cot a < 1 . 













6 


TRIGONOMETRY 


6. The functions of 30°, 45°, and 60° can be obtained as accurately 
as desired by elementary means. 


Example 1. Find the functions of 45°. 


Solution. 1. Construct an isosceles right triangle, 
unit long; each acute angle is 45°. 

2 . Since c 2 = a 2 + b 2 , c 2 = 1 + 1 = 2. 

3. By use of equations 1, page 3, with c = V2 , 
tan 45° = 1; cot 45° = 1; sec 45° = V2] 

sin 45° = — —^r\ csc 45° = V2; cos 45° = 

V 2 * 

Example 2. Find the functions of 30° and 60°. 


each of whose legs is 1 
B 



Solution. 1. Construct an equilateral triangle ABD each of whose sides is 
2 units long. Each angle of ABD is 60°. 


B 



A 6=1 C D 

Fig. 8 


2. Draw the bisector BC of angle ABD. Then, 
by plane geometry, BC is perpendicular to AD and C 
is the mid-point of AD. Hence, AC = 1 and angle 
ABC is 30°. 

3. In triangle ABC, 6 = 1 and c = 2. 

a 2 = c 2 — b 2 = 4 — 1 = 3; hence, a = Vs. 

4. By use of equations 1, page 3, from triangle 
ABC we obtain the results given for 30° and 60° in 
the following table. The irrational denominators in 
the table should be rationalize^ when the fractions 
are used. Thus, cot 60° = ^Vs. 


Angle 

SIN 

cos 

TAN 

COT 

SEC 

CSC 

46° 

1 

1 

1 

1 

V2 

V 2 


V2 

V 2 





CO 

0 

0 

1 

2 

Vs 

2 

1 

’ V3 

2 

Vs 

2 

60° 

V3 

2 

1 

2 

ICO 

> 

1 

Vs 

2 

2 

Vs 


The preceding table can be quickly re¬ 
produced and memorized by use of Figure 9 
from which we read the values of the func¬ 
tions. In the 45° triangle, the legs are 
equal. In the (30°, 60°) triangle the 
shortest leg is one-half of the hypotenuse. 



Fig. 9 























FUNCTIONS OF ACUTE ANGLES 


7 


7. Functions of particular angles may be obtained approximately by 
use of a diagram like Figure 10. 

Example 1. Find sin 50° and tan 50°. 

Solution. 1. Through 50° on the circu¬ 
lar arc, draw ABD. Construct BC and 
DF perpendicular to AF. By measuring 
lengths to the nearest half-unit, we find 
FD = 12.0 and CB =^7.5. 

2. From triangle ABC, 


. _ no CB 7.5 __ 

sm 60 - AS = To = • 76 • 


3. From triangle ADF, 


tan 50° = 


FD 12.0 


AF 


10 


= 1 . 20 . 


8 . Functions of complementary 
angles. If a and (3 are two acute 
angles such that a + j8 = 90°, then a 
and /3 are said to be complementary; 
either one is called the complement 
of the other angle. 

Illustration 1. The complement of 35° is (90° — 35°) or 55°. 



A i 


C 

Fig. 10 


If a is one acute angle of a right triangle ABC , then the other acute 
B angle, (3, is the complement of a. From Figure 11, 



a , Q u, 

sin a = - and cos p = -: 
c c 


or, sin a = cos j8 . 


Since (3 = 90° — a, hence we have proved that 
sin a = cos (90° — a). Similarly, the student should 
prove that the other formulas given below are true, 
sin a = cos (90° — a). sec a = esc (90° — a), 

cos a = sin (90° — a). esc a = sec (90° — a). [• (1) 

tan a = cot (90° - a). cot a = tan (90° - a). 


The names of the six trigonometric functions may be grouped in 
pairs as follows: sine and cosine; tangent and cotangent; secant and 
co secant. In each pair, either function may be referred to as the 
cofunction of the other one. Then, (1) may be summarized as follows: 
any function of a equals the cofunction of the complement of a. 

Illustration 2. sin 33° = cos 57°. tan 29° = cot 61°. esc 15 = sec 75 . 

Example 1. Express tan 37° 38' as a function of another acute angle. 

Solution. Since (90° — 37° 38') = 52° 22', hence tan 37° 38' = cot 52° 22'. 








































8 


TRIGONOMETRY 


EXERCISE 2 

Find the functions of 45° by use of an isosceles right triangle where each 
leg has the specified length: 

1 . Leg = 2. 2. Leg = 3. 3. Leg = 5. 4. Leg = V2. 5. Leg = h. 

Construct an equilateral triangle where each side has the specified length. 
Divide the triangle into two congruent right triangles and find the functions of 
30° and 60° from the resulting figure: 

6. Side = 4. 7. Side = 6. 8. Side = 8. 9. Side = 3. 

Without trigonometry, find the unknown sides of a right triangle, with 30° 
as one angle, under the given condition: 

10. Shortest leg is 3. 13. Hypotenuse is 12. 16. Hypotenuse is w. 

11. Shortest leg is 5. 14. Hypotenuse is 2 h. 17. Hypotenuse is 4 h. 

12. Hypotenuse is 8. 15. Shortest leg is k. 18. Shortest leg is 2 w. 

Tell the complement of each angle: 

19. 33°. 20. 47°. 21. 62°. 22. 58°. 23. 77°. 24. 85°. 


Express each function as a function of the complementary angle: 


25. sin 20°. 29. 

26. cos 15°. 30. 

27. tan 49°. 31. 

28. sec 32°. 32. 


cot 38°. 33. sec 

esc 12°. 34. tan 

cos 41°. 35. esc 

sin 73°. 36. sin 


81°. 37. tan 39°. 

54°. 38. sec 85°. 

67°. 39. cos 3°. 

58°. 40. sin 14°. 


Note. The names cosine, cotangent, and cosecant were originally introduced 
into trigonometry because of the existence of relations 1, page 7. Thus, in the 
Middle Ages, it was customary at first to refer to cosine, in Latin, as complementi 
sinus, meaning the sine of the complement. Eventually, complementi sinus was 
abbreviated to cosinus. 


Express as a function of an angle less than 45°: 

41. sin 79°. 44. esc 63° 10'. 47. cot 78°. 

42. cos 47°. 45. sec 80° 40'. 48. sin 57°. 

43. tan 85°. 46. cot 83° 20'. 49. cos 49°. 


50. sec 48° 50'. 

51. cot 71° 55'. 

52. tan 88° 6'. 


By use of Figure 10, draw the angle and find its sine, cosine, and tangent 
approximately to two decimal places by estimating lengths of lines: 

53. 10°. 54. 20°. 55. 30°. 56. 40°. 57. 45°. 58. 55°. 

Note. Inspect the triangles used for 10° and 50° in Figure 10; for each func¬ 
tion, notice how the change from 10° to 50° affects corresponding sides of the 
triangles and the value of the function. This inspection should illustrate the 
following facts, which will be discussed later: if a and y are acute angles and if 
a < y, then sin a < sin y, tan a < tan y, cos a > cos y, and cot a > cot y. 


FUNCTIONS OF ACUTE ANGLES 


9 


9. Significant digits. Let N be a number for which we have either 
an exact or an approximate expression in decimal form. Let us read 
digits from the left in N. Then, by definition, the significant digits, 
or figures, of N are those commencing with the first one not zero and 
ending with the last one definitely specified. In referring to significant 
digits, we do not mention any final zeros at the right if we are deal¬ 
ing with an exact value but we always mention final zeros in an approxi¬ 
mate value. 

Illustration 1. Notice that the significant digits of a number do not depend on 
the position of the decimal point. The significant digits of either 41.058 or .041058 
are (4, 1, 0, 5, 8). 

In referring to a place in a number, unless otherwise stated we shall 
mean a place where a significant digit stands and not necessarily a 
decimal place. 

Illustration 2. To say that N = 30.18 is a four-place approximation to N, or 
that 30.18 is the value of N correct to two decimal places , means that N differs 
from 30.18 by at most 5 units in the third decimal place. Hence, to the best 
of our knowledge, N lies between 30.175 and 30.185 inclusive. To say that 
N = 30.180 is a five-place approximation to N means that N lies between 30.1795 
and 30.1805. Notice the distinction in refinement between 30.18 and 30.180 as 
approximate values. 

To round off N to k places means to write the approximate value 
of N with k significant figures so that the error of the approximate 
value is not more than \ of a unit in its last place. 

Illustration 3. The seven-place approximation to tv is 7r = 3.141593. On 
rounding this off to five places we obtain tv = 3.1416. The two-place approxima¬ 
tion is tv = 3.1. 

In this book, unless otherwise specified or implied, any value given 
for a quantity is presumed to he an exact value. 

10. Four-place trigonometric table. In general, the decimal form 
of a function of an angle is an endless decimal. By use of advanced 
mathematics, the functions of an angle can be computed to as many 
decimal places as desired. In any table of values of the functions, 
the error in any entry is at most \ of a unit in the last place. 

Table VII is a four-place table of the functions of acute angles at 
intervals of 10'; with some exceptions, the entries are given to four 
significant digits. For angles at most equal to 45°, we read angles at 
the left and titles of columns at the top in the main part of each page 
of the table. For angles greater than 45°, we read titles at the bottom 
and angles at the right. 


10 TRIGONOMETRY 

Illustration 1. To find cot 5° 30', we look in the left-hand angle columns for 
5° 30' and then in the row of this angle we read the entry in the column headed 
cotangent at the top: cot 5° 30' = 10.39. To find sin 77° 20', we look for 77° 20' 
in the right-hand angle columns, and then in the row of 77° 20' we read the entry 
in the column labeled sine at the bottom: sin 77° 20' = .9757. 

Each entry in the function columns of Table VII is a function of 
some angle and, also, is the cofunction of the complementary angle. 
This feature, by which each entry serves a double purpose, is made 
possible by the relations of Section 8 . 

Illustration 2. We read in Table VII that .9757 = sin 77° 20' = cos 12 40 . 

Example 1. From Table VII, find a if cos a = .4173. 

Solution. We look for .4173 in the cosine columns in Table VII. We find 
.4173 on page 15 in the column with cosine at the bottom; hence we read the angle 
at the right: .4173 = cos 65° 20', or a = 65 20'. 

EXERCISE 3 


Find each function by use of Table VII: 


1 . 

sin 12°. 

6. cos 9° 20'. 

11. sec 18° 10'. 

16. 

cot 32° 50' 

2. 

tan 33°. 

7. sin 13° 

30'. 

12. sin 42° 40'. 

17. 

tan 5° 30'. 

3. 

cot 58°. 

8. tan 53 c 

’ 40'. 

13. tan 68° 50'. 

18. 

cos 43° 10' 

4. 

sec 64°. 

9. cot 85° 

20 '. 

14. cot 79° 20'. 

19. 

esc 62° O'. 

5. 

esc 73°. 

10. esc 73° 

50'. 

15. sec 88° 10'. 

20. 

sec 81° 10' 


Find the unknown acute angle a by use of Table VII: 


21. 

tana = .4074. 

27. esc a = 1.033. 

33. 

sin a = .6180. 

22. 

sin a = .1016. 

28. tana = 1.437. 

34. 

tana = 1.744. 

23. 

sec a = 1.167. 

29. cos a — .3035. 

35. 

cos a = .3502. 

24. 

cot a = 1.621. 

30. sec a = 1.932. 

36. 

tana = 21.47. 

25. 

cos a = .9261. 

31. cot a = .4841. 

37. 

sec a = 1.184. 

26. 

esc a = 4.134. 

32. cos a = .3584. 

38. 

esc a = 1.081. 


11. Interpolation in a four-place table. We shall obtain functions 
and angles from Table VII, which are not tabulated there, by a method 
called interpolation by use of proportional parts. In this method it is 
assumed that, for small changes in an angle a, the corresponding changes 
in the value of any one of the functions of a are proportional to the 
changes in a. This assumption, which is true only in an approximate 
sense, is called the principle of proportional parts. It can be proved 
that the method of interpolation based on this principle leads to re¬ 
sults which are sufficiently accurate for practical purposes. 


FUNCTIONS OF ACUTE ANGLES 


11 


Example 1. By use of Table VII, find sin 47° 43'. 

Solution. 1. We notice that 47° 43' is ^ of the way from 47° 40' to 47° 50', 
whose functions are given in the table. 

2. By the principle of proportional parts, an increase of 3' in the angle 47° 40' 
causes as much change in its sine as is caused by an increase of 10' in the angle. 
In other words, since 47° 43' is of the way from 47° 40' to 47° 50', hence sin 47° 43' 
is of the way from sin 47° 40' to sin 47° 50'. Therefore, 

sin 47° 43' = sin 47° 40' + .3(sin 47° 50' - sin 47° 40'). 


From table: 

sin 47° 40' = .73921 


Tabular difference is 


sin 47° 43' = ? 

20 

.7412 - .7392 = .0020. 

From table: 

sin 47° 50' = .7412_ 


.3(.0020) = .0006. 


3. Hence, sin 47° 43' = .7392 + .0006 = .7398. 

Comment. We call .0006 the proportional part of the tabular difference. In 
passing from .7392 to .7412 there is an increase of .0020; hence we added .0006. 

Example 2. Find cot 37° 27'. 

Solution. 1. Since 37° 27' is t t q of the way from 37° 20' to 37° 30', we assume 
that cot 37° 27' is ^ °f the way from cot 37° 20' to cot 37° 30'. 


From table: 

cot 37° 20' = 1.3111 


Tabular difference is 


cot 37° 27' = ? 

8 

1.311 - 1.303 = .008. 

From table: 

cot 37° 30' = 1.303J 


.7(8) = 5.6, or 6. 


2 . Hence, cot 37° 27' = 1.311 - .006 = 1.305. 

Comment. 1. We subtracted .006 because in passing from 1.311 to 1.303 we 
have a decrease in value. 

2. We changed from 5.6 to 6 in the third decimal place because the table en¬ 
tries we used are accurate only to three decimal places. 

3. We could read .7(8) = 5.6 directly from the column of tenths of 8 under 
Proportional Parts in the table. 

Example 3. Find a. if sin a. — .9254. 

Solution. 1. On page 15 of Table VII, in the column labeled sine at the bottom, 
we find the consecutive entries between which .9254 lies: 

.9250 = sin 67° 40' and .9261 = sin 67° 50'. 


2. Since .9254 is ^ of the way from .9250 to .9261, we assume that a is T 4 X of 
the way from 67° 40' to 67° 50'. 



1". T-9250 = sin 67° 40H 1 


^ = .36 = .4, to the nearest tenth. 

11 

4 1.9254 = sine J* 

10 ' 

x = .4(10') = 4'. 


.9261 = sin 67° 50' 


a = 67° 40' + .4(10') = 67° 44'. 









12 


TRIGONOMETRY 


Comment. 1. When using Table VII to find an unknown angle by interpo¬ 
lation, we agree to state the result only to the nearest minute. No greater re¬ 
finement is justified, because the unavoidable error, which may arise, frequently 
will be as large as one minute, although usually no larger. Hence, in Step 2 of 
Example 3, we computed to the nearest tenth, because later we were to multiply 
by 10'. 

2. In Step 2 of Example 3 we could obtain — A by merely inspecting the 
tenths of 11 in the column of proportional parts in Table VII. We read there that 

.3(11) = 3.3, or .3 = .4(11) = 4.4, or .4 = y~ 

Since 4 is nearer to 4.4 than to 3.3, hence fx is nearer to .4 than to .3. 


Example 4. Find a if cot a — 1.387. 

Solution. On page 16 in Table VII we find the consecutive cotangent entries 
1.393 and 1.385 between which 1.387 lies. Since 1.387 is f of the way from 1.393. 
to 1.385, hence we assume that a is f of the way from 35° 40' to 35° 50'. 


8 


r 1.393 = cot 35° 40' 
|_l-387 = cot a 


1.385 = cot 35° 50' 



| = .75 = .8 approximately, 
x = .8(10') = 8'. 
a = 35° 40' + .8(10') = 35° 48'. 


Comment. We found f = .75; we could call this either .7 or .8, to the nearest 
tenth. In this book, whenever such an ambiguity is met, we agree to choose 
the digit which makes the final result of the operation even in the last place. 


EXERCISE 4 


Find each function by use of Table VII: 


1 . 

tan 4° 43'. 8. 

cot 32° 38'. 

15. esc 65° 

39'. 

22. 

esc 76° 44'. 

2. 

tan 5° 17'. 9. 

sin 14° 24'. 

16. sec 53° 

13'. 

23. 

cos 35° 2'. 

3. 

sin 46° 52'. 10. 

tan 32° 36'. 

17. cot 77° 

16'. 

24. 

cot 67° 32', 

4. 

sin 47° 46'. 11. 

sec 45° 27'. 

18. cos 28° 

19'. 

25. 

esc 87° 23'. 

5. 

cot 27° 4'. 12. 

esc 56° 46'. 

19. sin 1° 11'. 

26. 

esc 4° 29'. 

6. 

cos 33° 15'. 13. 

sin 80° 17'. 

20. tan 6° 23'. 

27. 

tan 81° 53'. 

7. 

cos 24° 44'. 14. 

sin 28° 5'. 

21. sec 83° 

15'. 

28. 

tan 81° 54'. 

Find the acute angle a 

by use of Table VII: 




29. 

tan a = .0831. 

37. esc a 

= 1.265. 

45. 

tan a. = 1.026. 

30. 

sin a = .4955. 

38. sin a 

= .7967. 

46. 

sec 

a = 1.817. 

31. 

sin a = .3812. 

39. cos a 

= .7037. 

47. 

cos 

a = .4423. 

32. 

sec a = 1.128. 

40. tan a 

= 1.322. 

48. 

CSC 

a: = 3.462. 

33. 

cos a = .9381. 

41. sec a 

= 1.183. 

49. 

cot 

a = 2.612. 

34. 

cot a = 1.558. 

42. cos a 

= .9987. 

50. 

cos 

a = .7354. 

35. 

sec a — 1.506. 

43. sec a 

= 1.568. 

51. 

CSC 

a = 1.858. 

36. 

tan a = 1.031. 

44. cot a 

= .6720. 

52. 

sin 

a = .8698. 





FUNCTIONS OF ACUTE ANGLES 


13 


Find a, without interpolation, to the nearest 10' by use of Table VII: 

53. sin a = .2231. 54. tan a = 7.703. 55. cot a = 4.671. 

Hint for Problem 53. In Table VII, .2231 is nearer to .2221 than to .2250. 

56. cos a = .3437. 58. sin a = .6773. 60. esc a = 1.230. 

57. sec a = 2.018. 59. tana: = .6258. 61. cot a = .7601. 

12. Five-place values of the functions. In Table XI, for angles 
from 0° to 45°, minutes are read at the left margin and titles of 
columns at the top on the pages. For angles from 45° to 90°, minutes 
are read in the right margin and titles of columns at the bottom. 

Illustration 1. To find sin 67° 27', we go to page 96 in Table XI, in the sine 
column for 67° and in the row labeled 27' at the right: sin 67° 27' = .92355. 

Illustration 2. On page 101 in Table XI we find tan 32° 13' = .63014. 

Example 1. Find the acute angle a if sin a = .57071. 

Solution. We search for .57071 in the sine-cosine columns of Table XI: in the 
sine column for 34° we find sin 34° 48' = .57071, or a = 34° 48'. 

Example 2. Find the acute angle a if cot a = .68942. 

Solution. We search for .68942 in the tangent-cotangent columns of Table IX: 
in the cotangent column for 55° we find cot 55° 25' = .68942, or a = 55° 25'. 

★ Example 3.* Find cos 57° 33.6' by interpolation in Table IX. 

Solution. The desired cosine is of the way from cos 57° 33' to cos 57° 34'. 


From table: cos 57° 33' = .53656”1 


x = .6(24) = 14.4. 

cos 57° 33.6' = ? J * 

24 

cos 57° 33.6' = .53656 - .00014; 

From table: cos 57° 34' = .53632 


cos 57° 33.6' = .53642. 


★ Example 4. Find a by interpolation if sin a — .03914. 

Solution. In the sine column for 2° in Table XI we find the entries .03897 
and .03926 between which .03914 lies. 



" T.03897 = sin2°14H ~ 

L-03914 = sin a 

-59 = .6, approximately. 

29 

1' Hence, x = .6(1') = .6'. 


.03926 = sin 2° 15' 

a = 2° 14' + .6' = 2° 14.6'. 


Comment. When using Table XI to find an unknown angle, we agree to state 
the result to the nearest tenth of a minute; no greater refinement is justified be¬ 
cause the unavoidable error, which may arise, frequently will be as large as .1' 
although usually no larger. 

* The omission of the supplementary material marked with a star ★ will not disturb 
the continuity of the text. 









14 


TRIGONOMETRY 


EXERCISE 5 


Find each function, or the unknown angle a, by use of Table XI: 


1. 

sin 3° 

ir. 6. 

cos 14 

0 40'. 11. cot 83° 

59'. 

16. cos 52° 37'. 

2. 

cos 5° 

27'. 7. 

CSC 72' 

3 18'. 12. sec 15° 

8 '. 

17. tan 28° 17'. 

3. 

sec 83' 

3 12'. 8. 

sin 43‘ 

5 5'. 13. esc 41° 

51'. 

18. sec 73° 19'. 

4. 

cot 53 

0 19'. 9. 

cos 31 

° 15'. 14. cos 38° 

35'. 

19. esc 51° 38'. 

5. 

tan 32 

° 49'. 10. 

tan 49 

° 2'. 15. sin 54° 

28'. 

20. sin 89° 59'. 

21. 

sin a 

= .53263. 

27. 

sin a = .30431. 

33. 

tan a = .20557. 

22. 

cos a 

= .57238. 

28. 

tan a = 2.8344. 

34. 

cot a = 2.8057. 

23. 

tan a 

= .02677. 

29. 

cos a = .43418. 

35. 

esc a = 1.0811. 

24. 

cot a 

= 101.11. 

30. 

cot a = .29432. 

36. 

sin a = .85234. 

25. 

sec a 

= 1.0376. 

31. 

sin a = .12995. 

37. 

cos a = .27564. 

26. 

esc a 

= 1.9230. 

32. 

sec a = 1.2633. 

38. 

sec a = 1.6553. 

Without interpolating, by inspection of Table XI find the acute angle a cor¬ 
rect to the nearest minute: 

39. 

sin a 

= .29245. 

42. 

cot a = 6.9592. 

45. 

cos a = .68125. 

40. 

cos a 

= .34103. 

43. 

cos a = .83824. 

46. 

sin a = .03786. 

41. 

tan a 

= 3.2869. 

44. 

sin a = .44192. 

47. 

tan a = .62718. 

icFind each function by interpolation in Table XI : 



48. 

sin 32‘ 

3 13.6'. 

53. 

sin 31° 51.7'. 

58. 

sin 57° 39.9'. 

49. 

tan 34 

° 32.3'. 

54. 

cos 81° 28.6'. 

59. 

cos 71° 11.5'. 

50. 

cot 76 

° 35.4'. 

55. 

sec 15° 8.2'. 

60. 

cot 43° 36.2'. 

51. 

cos 54 

° 40.7'. 

56. 

esc 23° 19.1'. 

61. 

esc 27° 47.6'. 

52. 

tan 21 

° 18.8'. 

57. 

cot 47° 17.3'. 

62. 

sec 49° 53.8'. 

t 

+Find the acute angle a by interpolation in Table XI : 


63. 

tan a 

= .21902. 

68. 

cos a = .87241. 

73. 

sin a = .75922. 

64. 

sin a 

= .70037. 

69. 

cot a = .54878. 

74. 

cos a = .53375. 

65. 

cos a 

= .93868. 

70. 

sec a = 3.5056. 

75. 

cot a = .07884. 

66. 

cot a 

= 2.4286. 

71. 

sec a = 1.0377. 

76. 

esc a = 1.0394. 

67. 

sin a 

= .21567. 

72. 

tan a = 4.4389. 

77. 

esc a = 5.1738. 


13. Solution of a right triangle. A right triangle has six parts, 

three angles and three sides, of which one angle is 90°. By means of 
trigonometry, if two sides, or an acute angle and a side, of a right tri¬ 
angle are given,* we can compute the unknown parts of the triangle; 
this computation is called the solution of the triangle. 

* Recall that then it is possible to construct the triangle by plane geometry. 


FUNCTIONS OF ACUTE ANGLES 


15 


Recall the following formulas for the right triangle in Figure 12. 


(1) a 2 + & 2 = c 2 . 

(3) sin a = ^ = cos /3. 

(5) tan a = ^ = cot 0. 
b 

Q 

(7) sec a = ^ = esc /3. 


(2) a + 0 = 90°. 

(4) cos a = ^ = sin /3. 

(6) cot a = ^ = tan 0. 

Q 

(8) esc a = - = sec /S. 





AbC 
Fig. 12 


From (3) and (4) we obtain the following useful formulas: 

a = c sin a; b = c cos a; (9) 

or, the leg opposite ol equals the hypotenuse times sin ot, and the leg ad¬ 
jacent to a equals the hypotenuse times cos ot. We shall avoid using 
(1) until we have met logarithms. As an aid to accuracy, we shall 
usually employ (3) or (4) in preference to (7) or (8), in finding un¬ 
known angles. Formulas 7 and 8 are useful in avoiding division 
when finding an unknown side c. 


Suggestions for solving a triangle. 


I. Make a preliminary sketch roughly to scale for the data. 

II. To find any unknown part, employ a formula which involves it but no 
other unknown. When possible, select a formula which leads to multiplica¬ 
tion instead of to division, to simplify the arithmetic. 

III. As a rough check on the solution, compare the results with the pre¬ 
liminary sketch. For a more refined check, substitute in any one of formulas 
3 to 8 which was not used in the solution and which involves as many as possible 
of the computed elements. 

IV. In the arithmetic, round off all lengths, quotients, and products to four 

or to five significant digits, and all angles to minutes or to tenths of a minute, 
according as four-place or five-place tables are used. b 


Example 1. Solve triangle ABC if b = 250 and c 
Solution. 1. Outline. To obtain (3, csc/3 = -• 
From a + p = 90°, a = 90° - j8. 

From cot ft = y a = b cot (3. 


Check formula: a = c sin a. 

2. Computation. C3C (3 = = 2.872. 

By interpolation in Table VII, /3 = 20° 23'. 

Hence, a = 90° - 20° 23' == 69° 37'. 



a = 250 cot 20° 23' = 250(2.692) = 673.0. (Using Table VII) 


Check, c sin a = 718 sin 69° 37' = 718(.9374) = 673.1. = 673.0. 

The check is satisfactory; the two sides of a = c sin a differ by only .1. 






16 


TRIGONOMETRY 


Example 2. Solve triangle ABC if a = 30.5 and /3 = 32° 10'. 
Solution. 1. Outline. From a + P = 90°, a — 90° — j3. 

From - = tan P, b = a tan /3; 

a 

From - = sec P, c = a sec/3. 

a 

Check formula: & = c cos a. 

EXERCISE 6 

Solve triangle ABC of Figure 12 by use of Table VII: 


1. 

a = 

50; a = 32° 20'. 

11. 

a 

= 

2.3; b = 

1.25. 

2. 

b = 

75; a = 68° 40'. 

12. 

c 

= 

.43; a = 

.262. 

3. 

c = 

12.5; P = 13° 20'. 

13. 

c 

= 

.73; b = 

.425. 

4. 

c = 

15; a = 56° 30'. 

14. 

a 

= 

89.6; b = 

25.4. 

5. 

c = 

7.5; a = 5.083. 

15. 

a 

= 

.013; p = 

= 52° 11'. 

6. 

b = 

85.22; a = 65. 

16. 

b 

= 

.38; a = 

41° 59'. 

7. 

a 

31.5; a = 29° 32'. 

17. 

c 

= 

1.6; p = 

CO 

0 

£>- 

°0 

8. 

c = 

1.4; P = 73° 47'. 

18. 

b 

= 

59,000; a 

= 32,500. 

9. 

c = 

.58; a = .259. 

19. 

b 

= 

45,000; c 

= 86,000. 

10. 

b = 

.24; c — .485. 

20. 

a 

= 

.135; a = 

= 79° 28'. 


Problems 21 to 30. Solve Problems 11 to 20, respectively, by use of 
Table XI without interpolation, rounding off angles to the nearest minute 
and sides to four significant digits. 

Note. Use of Table XI in the fashion specified above gives about the same 
accuracy as is obtainable by interpolation in Table VII. 

+Solve Problems 31 to 40 by use of Table XI, obtaining angles to tenths of a 
minute and sides to five significant digits: 

31. a = 35; a = 21° 16.7'. 33. c = 13.6; p = 15° 45.7'. 

32. b = 14; a = 42° 31.3'. 34. c = .88; a = 63° 18.4'. 

Problems 35 to 40. Use the data of Problems 9 to 14, respectively. 

14. Applications. In any geometrical application of trigonometry, 
perform the following steps before computing. 

I. Construct a figure roughly to scale. 

II. Introduce single letters to represent unknown angles or lengths. 

III. Outline the solution, specifying each triangle and formula to be used 
and solve each formula for the quantity to be obtained from it. 


17 


FUNCTIONS OF ACUTE ANGLES 

In either diagram in Figure 14, 0 is a point from which we sight an 
object at C and OH is a horizontal line in the same vertical plane as 
C. Then, the angle COH between the line of sight to C and the hori¬ 
zontal line is called the angle of elevation of C or the angle of de¬ 
pression of C, as seen from 0, according as C is above 0 or below 0. 



Fig. 14 


Example 1. From a cliff, 700 feet above a horizontal plane, the angle of 
depression of a church in the plane is 38° 27'. Find Ar 
the distance from the cliff to the church. 

Outline of solution. 1. In Figure 15, C represents 
the church. The angle of depression is CAH, repre¬ 
sented by 9 (called theta). We desire the value of x. 

2. In triangle CKA, y = 700 and 9 = 38° 27'. 


3. Since cot 9 = hence, 

y 


x = y cot 



Fig. 15 


EXERCISE 7 * 

Solve by use of four-place tables unless otherwise directed: 

1. Find the length of the horizontal shadow of a tree 95 feet tall when 
the angle of elevation of the sun is 73° 17'. 

2. Find the angle of elevation of the sun if a steeple 185 feet tall casts a 
horizontal shadow 80 feet long. 

3. An artillery observer in a captive balloon 2700 feet above his guns 
observes that the angle of depression of an enemy’s fort is 27° 56'. Find 
the distance from the guns to the fort, if they lie at the same elevation. 

4. A guy wire 70 feet long is stretched from the ground to the top of a 
telephone pole 50 feet high. Find the angle between the wire and the pole. 

5. From a mountain top 4000 feet above a city, its angle of depression 
is 16° 45'. Find the air-line distance from the mountain top to the city. 

* These problems may be reserved until after logarithms are introduced. 












18 


TRIGONOMETRY 


6. From an airplane, flying 7000 feet above the ground, the angle of 
depression of a landing field is 19° 32'. Find the air-line distance from the 
plane to the field. 

7. Find the height of the Empire State Building in New York City if 
the angle of elevation of its top is 61° 37' when seen from a point on the 
street level 675.4 feet from the building. 

8. A guy wire 120 feet long runs from the top of a pole to a point 65 
feet from its foot. How high is the pole? 

9. The dimensions of a rectangle are 38' by 17'. Find the length of a 
diagonal and the acute angle between it and a short side. 

10. Find the height of a flagpole whose horizontal shadow is 70 feet 
long when the elevation of the sun is 58° 33'. 

11. From a point on one rim of a valley 600 feet deep with vertical sides, 
the angle of depression of an object directly opposite on the valley floor is 
53° 28'. How wide is the valley? 

12. The largest tree in California is the General Sherman tree in the 
Sequoia National Park. At a point 185 feet from the tree, at the same 
elevation as its foot, the angle of elevation of the top of the tree is 55° 49'. 
How tall is the tree? 

13. How high does an airplane rise in flying 8000 feet upward along a 
straight path inclined 28° 47' from the horizontal? 

14. An inclined ramp leading into a garage is 130 feet long and rises 
38 feet. Find the inclination of the ramp to the horizontal. 

15. A painter desires to reach a window 40 feet above the ground. Find 
the length of the shortest ladder he can use if it must not incline more than 
78° from the horizontal. 

16. From a cliff 150 feet above a lake, we see a boat sailing directly 
toward us. The angle of depression of the boat is seen to be 5° 7' and 
then later is 11° 18'. Find the distance the boat sailed between these 
observations. 

17. From a mountain top 5000 feet above a horizontal plane, we ob¬ 
serve two villages in the plane due east of us, whose angles of depression are 
8° 38' and 5° 46'. How far apart are the villages? Use Table XI. 

18. On a 3% railroad grade, at what angle are the rails inclined to the 
horizontal, and how far does one rise in traveling upward 5000 feet measured 
along the rails? Use Table XI. 

Hint. To say that the grade is 3% means that the tracks rise 3 feet for each 
100 feet of horizontal distance gained. 

19. If the grade of a railroad is 6.75%, how far must one travel along 
the rails to rise 500 feet? Use Table XI. 

20. The light on the mast of a boat is 64 feet above water level. From 
an observing instrument^ feet above water level on the shore, the angle of 
elevation of the light is 3° 14'. How far away is the boat? 


FUNCTIONS OF ACUTE ANGLES 


19 


REVIEW EXERCISE 8 

Construct the acute angle a and find its other functions without a table: 

1. cos a = 2. sin a = -fa. 3. tan a — $. 4. cot a = 

Suppose that (3 is the complement of a. By mere inspection tell the value of 
some function of 0 and of some other function of a, under the given condition. 

5. sin a = 7. tan a = 9. sec a = 8. 11. sin a = f. 

6. cos a = bi- 8. cot a — 10. esc a = 4. 12. cos a = f. 

13. Determine the functions of 30° and 60° by use of an equilateral tri¬ 
angle each of whose sides is 8 units long. 

14. Determine the functions of 45° by use of an isosceles triangle each of 
whose sides is 6 units long. 

15. By use of a right triangle, prove that (1) the secant of any acute angle 
equals the cosecant of its complement; (2) the cotangent of any acute angle 
equals the tangent of its complement. 

Express each function as a function of some other acute angle: 

16. sin 47°. 18. tan 59°. 20. esc 81°. 22. cos 53°. 

17. cos 63°. 19. sec 73°. 21. cot 52°. 23. sin 79°. 

Use Table VII in the problems below unless otherwise directed. 

Solve triangle ABC completely except where only a single part is asked for: 

24. a - 38; c = 60; find a. 26. c = 80; a = 32° 27'; find b. 

25. a = 50; b = 67; find 0. 27. c = 60; 0 = 43° 26'; find a. 

28. b = 28; c = 40. 29. c = 25; a = 43° 27'. 30. a = 60; 0 = 57° 19'. 

31. How tall is a chimney whose horizontal shadow is 90 feet long when 
the angle of elevation of the sun is 67° 42'? 

32. An aviator is flying due north at an elevation of 3500 feet above two 
towns, one due north and the other due south of him in the same horizontal 
plane. If the angles of depression of these towns as seen by the aviator are 
19° 35' and 22° 46', respectively, find the distance between the towns. 

33. A tower stands on top of a cliff above a horizontal plane. At a point 
in the plane 750 yards from the cliff, the angles of elevation of the top and 
the bottom of the tower are 47° 20' and 45° 12'. How high is the tower? 

•kFind each function, or unknown angle, by interpolation in Table XI: 

34. sin 48° 26.7'. 37. esc 17° 52.8'. 40.* sin 35° 16' 42". 

35. cos 23° 18.4'. 38. cot 78° 23.6'. 41.* cos 23° 48' 50". 

36. sin a = .19526. 39. cos a = .98577. 42. tan a = .20653. 

* Express 42", or 50", approximately as tenths of a minute using 1' = 60". 


CHAPTER II 


LOGARITHMS 

15. Simplification of computation. We shall define certain aux¬ 
iliary numbers called logarithms. These will permit us to change the 
operations of multiplication, division, raising to a power, and extrac¬ 
tion of a root to the easier operations of addition, subtraction, multi¬ 
plication, and division, respectively, applied to the proper logarithms. 

16. Rational and irrational numbers. A number which can be 
expressed as a fraction u/v, where u and v are integers, is called a 
rational number. A real * number which is not a rational number is 
called an irrational number. 

Illustration 1. 4, — 5, 0, and ™ are rational numbers, tv, V3, and 'v'S are 
irrational numbers. 

17. Exponents. We call a m the rath power of the base a, and ra the 
exponent of the power. We recall the following relations, where ra 
and n represent positive integers and p is any rational number. 


a m = a ■ a ■ a ■ ■ a 

(m factors ). 

(1) 

0 ° = 1 . 


(2) 

an = v 7 a m = (v / a) m . 

(n not even if a < 0) 

(3) 

ft 

1 

II 


(4) 


In elementary algebra the following theorems, called index laws, 
are proved to hold if the exponents are rational numbers. 

I. Law of exponents for multiplication: a x a y = a x+y . 

a x 

II. Law of exponents for division: — = a xy . 

III. Law for finding a power of a power: {a x ) k = a kx . 

Illustration 1. a* = a ■ a ■ a • a. 16° =1- 8® = ^8 = 2. 

a$ = VT 3 . 16* = (vT6) 3 = 2 3 = 8. ^ = a 11 . = 5 ~ 3 . 

t a 6 5 d 

= 2V 6 . a 3 a 5 = a 8 . (x 3 ) 2 = x 6 . 

We shall use irrational numbers as exponents, but a logical foun¬ 
dation for their use is beyond the scope of this text. Hence, without 

* In this book, unless otherwise stated, any number referred to is a real number. 

20 


LOGARITHMS 


21 


discussion, we shall assume that irrational powers have meaning 
intuitionally, and that the preceding index laws hold if the exponents 
are any real numbers, if the base is positive. 

18. Logarithms. In the following definition, a represents a positive 
number, not 1, and N is any positive* number. 

Definition I. The logarithm of a number N to the base a is the ex¬ 
ponent of the power to which a must be raised to obtain N. 

In other words, if N = a x , then x is the logarithm of N to the base 
a. To abbreviate “ the logarithm of N to the base a ” we write “ log a N.” 
Then, by Definition I, the following equations state the same fact: 

N = a x \ x = loga N. (1) 

Illustration 1. Since 1000 = 10 3 , hence log 10 1000 = 3. 

If logs N = 3, then N = 6 3 = 216. 

If logas N - i, then N = 25s = V25 = 5. 

Since a 0 = 1 and a 1 = a, the following relations are true for every 
value of the base a. 

log a 1 = 0. log a a = 1- (2) 

Example 1. Find logs 

Solution. Express ^ as a power of the base, 5: 

ak ~h' or 6^ = 5 ~‘- He " ce ' log ‘6k"~ 3 ' 

★ Example 2. Find the unknown base if log a 25 = — 2. 

Solution. 1. By equations 1, a~ 2 = 25. 

2. Hence, ^ = 25; ^ = a 2 . Therefore, a = i- 

Note 1. We do not allow a = 1 as a base because every power of 1 is 1 and 
hence no number except 1 could have a logarithm to the base 1. If a negative 
base were used, the logarithms of most numbers would be imaginary numbers. 
Hence, we took a > 0 and s^l. 

We shall assume the truth of the following facts which are proved 
in advanced mathematics: 

1. For every positive number N, there exists one and only one real number x 
such that N — a x . In other words, every positive number N has one and only 
one real logarithm to the base a. 

2. If M < N and a > 1, then log a M < log a N. 

* In this book, if we mention the logarithm of a number N, we mean a positive 
number N. If N < 0, log a N may be defined as an imaginary number. 


22 


TRIGONOMETRY 


EXERCISE 9 


Express as a fraction: 


1. a" 5 . 2. b~ 3 . 3. 2 -4 . 4. 3~ 4 . 5. 10" 2 . 6. 10" 3 . 

Express by use of a fractional exponent: 

7. Va. 8. </b. 9. 10. v^. 11. ^10. 12. ^10. 


By use of equations 1, write an equivalent logarithmic equation: 


13. TV = 3 6 . 

14. TV = 5 2 . 

15. N = 10 4 . 

16. TV = 10~ 3 . 

17. N = 6- 2 . 

18. TV = 30 -4 . 


19. TV = 5i 

20. TV = 6i 

21. TV = 10*. 

22. TV = lOi 

23. TV = 10- 36 . 

24. TV = 10*>. 


25. TV = 10-- 4 . 

26. TV = 10-- 6 . 

27. 49 = 7 2 . 

28. 64 = 2®. 

29. 64 = 8 2 . 

30. 27 = 3 3 . 


31. 125 = 5 3 . 

32. 625 = 5 4 . 

33. 64 = 4 3 . 

34. * = 5“ 2 . 

35. * = 2 -5 . 

36. = IQ" 2 . 


Find the number whose logarithm is given: 

42. log 7 TV = 3. 


37. logio TV = 5. 

38. logs TV = 3. 

39. logs TV = 2. 

40. logs TV = 2. 

41. log 8 TV = 3. 

52. log 4 TV = i 53. 


43. logxo TV = 4. 

44. logio TV = 0. 

45. logio TV = 1. 

46. log 9 TV = 1. 


47. logo TV = 1. 

48. logio TV = — 1. 

49. logs TV = 0. 

50. logio TV = — 2. 


51. logs TV = - 3. 
logs TV = i 54. logs TV = i 55. logi* TV = 


Find the following logarithms: 

56. logs 36. 57. log 4 64. 58. log 2 32. 59. log 4 2. 

Hint. Express the number, whose logarithm is desired, as a power of the base. 

60. logio 100. 63. logs 216. 66. log 9 3. 69. logs 

61. log 2 16. 64. logioo 10,000. 67. logioo 10. 70. log 3 *. 

62. logs 27. 65. log 3 81. 68. logs i. 71. logio .0001. 


SUPPLEMENTARY PROBLEMS 


Find a , TV, or x, whichever is not given: 


72. log a 16 = 2. 

73. logo 125 = 3. 

74. log a 625 = 4. 

75. log a 1000 = 3, 

76. loga 9 = 

77. log a 3 = 


78. loga 10 = b 

79. loga 2 = |. 

80. loga* = - 1. 

81. loga .001 = - 3. 

82. logsi TV = f. 

83. logioo TV = -§■. 


84. log.oi TV = — ■§. 

85. logio .1 = x. 

86. logio x = b 

87. loga .0001 = - 2. 

88. log 626 25 = x. 

89. loga 8 = - f. 


LOGARITHMS 


23 


19. Properties of logarithms useful in computation. 

Property I. The logarithm of a product equals the sum of the log¬ 
arithms of its factors. For instance, 

loga MN = loga M + log a N. (1) 

Illustration 1. Logio (897) (596) = logio 897 + logio 596. 

Proof. Let x = log a M, and y = log a N. Then, 

M = a x , and N = a y . (Definition of a logarithm) 

MN = a x a v = a x+y . (A law of exponents) 

Therefore, by Definition I, log a MN = x + y = log a M + log a N. 

Note 1. By use of (1) we can prove Property I for a product of any number of 
factors. Thus, since MNP = ( MN)(P ), 

loga MNP = loga MN + logo P = log a M + logo N + logo P. 

Property II. The logarithm of a quotient equals the logarithm of the 
dividend minus the logarithm of the divisor: 

M 

loga ^ = loga M - loga N. (2) 

89 

Illustration 2. Logio gy = logio 89 — logio 57. 

Proof. Let x = logo M, and y = log a N, then, M = a x ; N = a y . 
Therefore, ^ ^ = a x ~ y . (A law of exponents) 

Hence, log a ™ = x - y = loga M - log a N. (Definition I) 

Property III. The logarithm of the kth power of a number N equals 
k times the logarithm of N: 

loga N k = k log a N. (3) 

Proof. Let x = log a N. Then, by Definition I, N = a x . 

Therefore, N k = {a x ) k = a kx . (A law of exponents) 

Hence, by Definition I, log a N k = kx = k log a N . 

Illustration 3. logo 6 8 - 5 = 8.5 logo 6. 

If h is any integer, then ^N = N». Hence, by use of Property III with 
k = r ’ we obtain 

loga ^ loga N. (4) 

loga Vn = i loga N; loga ^25 = £ loga 25. 


Illustration 4. 


24 


TRIGONOMETRY 


★EXERCISE 10 


Find the logarithm of each number to the base 10 

, given 

that 



log xo 

2 = 

.3010; 

logio 3 

= .4771; 

logio 7 

= .8451; 

logio 

17 = 

= 1.2304. 

Illustration, logio 20 = 

logio (10 • 

2) = logio 

10 + 

logio 2 

= 1 

+ .3010. 

1. 

6. 

6. 


11. 


16. 

8. 


21. 


2. 

34. 

7. 

%• 

12. 

A* 

17. 

V~7. 


22. 

</2l. 

3. 

30. 

8. 


13. 

9. 

18. 

Vbl. 


23. 

VI 

4. 

70. 

9. 

ion. 

14. 

49. 

19. 

i 


24. 

V&. 

6. 

i- 

10. 

rr. 

15. 

27. 

20. 

h 


25. 

V&. 


Prove by the method used in proving Properties I and II: 

26. logo MNP = loga M + loga N + log a P. 

27. loga l(MP) + Q] = loga M + loga P ~ loga Q. 

20. Logarithms to the base 10 are called common, or Briggs, log¬ 
arithms. Hereafter, unless otherwise stated, when we mention a log¬ 
arithm we shall mean a common logarithm. Instead of writing logio N 
for the common logarithm of N, we shall omit the 10 and write 
merely log N. The following common logarithms will be useful later. 
The student should obtain these logarithms by use of Definition I. 


Number 

.0001 

.001 

.01 

.1 

1 

10 

100 

1000 

10,000 

100,000 

Logarithm 

- 4 

- 3 

- 2 

- 1 

0 

1 

2 

3 

4 

5 


Note 1. Common logarithms are the most convenient logarithms for com¬ 
putational purposes. The only other variety used appreciably is the system of 
natural, or Naperian, logarithms, in which the base is a certain irrational number 
denoted by e where e = 2.71828, approximately. Naperian logarithms are use¬ 
ful for theoretical reasons. 

Every real number, and hence every logarithm, can be written as the 
sum of an integer and a decimal fraction, which is positive or zero, and 
less than 1. After log N is written in this way, we call the integer 
the characteristic, and the fraction the mantissa, of log N: 
log N = (an integer) + (a fraction, ^ 0, < 1). 1 
log N = characteristic + mantissa. J 

From (1) it is seen that the characteristic of log N is negative if and 
only if log N itself is negative. 

Illustration 1. Log N is an integer when and only when N is an integral 
power of 10. Thus, if 100 < N < 1000, then log 100 < log N < log 1000; or, 
2 < log N < 3; hence, log N = 2 + (a fraction, > 0, < 1). 














LOGARITHMS 


25 


Illustration 2. The following logarithms were obtained by later methods. 


Logarithm 

Characteristic 

Mantissa 

log 300 = 

2.47712 = 

2 + -47712 

2 

.47712 

log .0385 = 

- 1.41454 = 

- 2 + .58546 

- 2 

.58546 

log 50 = 

1.69897 = 

1 + .69897 

1 

.69897 

log .001 = 

- 3.00000 = 

- 3 + .00000 

- 3 

.00000 

log 6.5 = 

0.81291 = 

0 + .81291 

0 

.81291 


Note 2. Logarithms were invented by a Scotchman, John Napier, Baron of 
Merchiston (1550-1617). His original logarithms were not the same as those now 
called Naperian logarithms. Common logarithms were invented by an English¬ 
man, Henry Briggs (1556-1631), who was aided in his invention by Napier. 


21. Properties of the characteristic and the mantissa. Hereafter 
in this chapter we shall assume that any number referred to is ex¬ 
pressed in decimal form. 


Illustration 1. All numbers whose logarithms are given below have the same 
significant digits, (3, 8, 0, 4). To obtain the logarithms, log 3.804 was first found 
from a table to be discussed later; the other logarithms were then obtained by 
use of Properties I and II, page 23. 


log 

380.4 

= log 100(3.804) 

= log 

100 + 

log 3.804 = 

log 

38.04 

= log 10(3.804) 

= log 

10 + 

log 3.804 = 

log 

3.804 

= .5802 



= 

log 

.3804 

, 3.804 

= 1 °S 10 

= log 

3.804 

— log 10 = 

log 

.03804 

. 3.804 

= l0g 100 

= log 

3.804 

— log 100 = 


2 + .5802 
1 + .5802 
0 + .5802 

- 1 + .5802 


- 2 + .5802. 


Similarly, if N is any number whose significant digits are (3, 8, 0, 4), then N equals 
3.804 multiplied, or else divided, by a positive integral power of 10; hence, it 
follows as above that .5802 is the mantissa of log N. 


In Illustration 1 we met special cases of the following theorems. 

Theorem I. The mantissa of log N depends only on the sequence 
of significant digits in N. That is, if two numbers differ only in the 
position of the decimal point, their logarithms have the same mantissa. 

Theorem II. When fV > 1, the characteristic of log N is an integer, 
positive or zero, which is one less than the number of digits in N to 
the left of the decimal point. 

Theorem III. If N < 1, the characteristic of log N is a negative 
integer; if the first significant digit of N is in the fcth decimal place, 
then — k is the characteristic of log N. 












26 


TRIGONOMETRY 


Illustration 2. By use of Theorems II and III, we find the characteristic of 
log N by merely inspecting N. Thus, by Theorem III, the characteristic of 
log .00039 is — 4 because “3” is in the 4th decimal place. By Theorem II, the 
characteristic of log 15,786 is 4, because there are five digits to the left of the 
decimal point (which is understood after 6). 

EXERCISE 11 

Each number is the logarithm of some number N. State the characteristic 
and the mantissa of log N. 

1. 3.95. 2. 4.86. 3. - 1.316. 4. - 2.165. 5. .587. 6. - .627. 

Find the characteristic of the logarithm of the number: 

7. 2950. 9. .0036. 11. 1.47. 13. 63,567. 15. .00009. 

8. 654. 10. 6.27. 12. .0139. 14. .00062. 16. 580,000. 

Find the common logarithm of each number by use of Definition I, and tell the 
characteristic and the mantissa of the logarithm: 

17. 1,000,000. 19. .001. 21. 10. 23. 10i 25. 10 1 - 67 . 

18. .00001. 20. 100. 22. 1. 24. VlO. 26. 10“ 2 - 7 . 

Qiven that log 4.57 = .6599, find each logarithm: 

27. log 45.7. 28. log 45,700. 29. log .0457. 30. log .000457. 

Find the characteristic of log N under the given condition: 

31. 1 ^ N < 10. 33. 10 ^ N < 100. 35. .1 ^ N < 1. 

32. 100 ^ N < 1000. 34. 1000 ^ N < 10,000. 36. .01 ^ N < .1. 

22. Standard form for a negative logarithm. Hereafter, for con¬ 
venience in computation, if the characteristic of log N is negative, 
- k, change it to the equivalent value 

[(10 — k) — 10], or [(20 — k) — 20], etc. 

Illustration 1. Given that log .000843 = — 4 + .9258, we change — 4 to 
(6 — 10) and write 

log .000843 = - 4 + .9258 = (6 - 10) + .9258 = 6.9258 - 10. 


The characteristics of the logarithms in the following table are ob¬ 
tained by Theorem III; the mantissas are identical, by Theorem I. 


1st Signip. Digit in 

Illustration 

Log N 

Standard Form 

1st decimal place 
2nd decimal place 
3rd decimal place 
6th decimal place 

N = .843 

N = .0843 

N = .00843 

N = .00000843 

- 1 + .9258 = 9.9258 - 10 

- 2 + .9258 = 8.9258 - 10 

- 3 + .9258 = 7.9258 - 10 

- 6 + .9258 = 4.9258 - 10 










LOGARITHMS 


27 


23. Tables of logarithms. Mantissas can be computed by use of 
advanced mathematics, and, except in special cases, are unending 
decimal fractions. Computed mantissas are found in tables of loga¬ 
rithms, also called tables of mantissas. 

Illustration 1. The mantissa for log 10705 is .029586671630457, to fifteen 
decimal places. In a five-place table of logarithms this mantissa would be re¬ 
corded correct to five decimal places, giving .02959. 

Definition II. A number N is called the antilogarithm of L if 
log N = L. We write N = antilog L. 

Illustration 1. Since log 1000 = 3, hence 1000 = antilog 3. 

Given that log 5 = .6990, hence 5 = antilog .6990. 

To find antilog 1.632, means to find the number N such that log N — 1.632. 

We find logarithms and also antilogarithms by use of tables of 
logarithms. 

Note 1. The first table of common logarithms was published by Henry 
Briggs in 1624. 

24. Four-place logarithms.* Table V gives the mantissa of log N 
correct to four decimal places, if N is any number with at most three 
significant digits. 

Example 1. Find log .000843 from Table V. 

Solution. 1. By Theorem III, the characteristic of log .000843 is — 4. 

2. To obtain the mantissa for 843 from Table V, find the first two digits “84” 
in the column headed N; in the row of “84,” in the column for “3,” we find that 
the mantissa for 843 is .9258 (we supply the decimal point for the entry). 

3. Hence, log .000843 = - 4 + .9258 = 6.9258 - 10. 

Example 2. Find N by use of Table V if log N = 7.6064 — 10. 

Solution. 1. We find the significant digits from Table Y. The mantissa is 
.6064, which is found in Table V as the mantissa corresponding to 404; this is 
the significant part of N. 

2. The characteristic of log N is (7 — 10), or — 3. Hence, by Theorem III, 
the first significant digit of N is in the third decimal place: N = .00404. 

Comment. We may say that N = antilog (7.6064 — 10) = .00404. 

Note 1. If 1 N < 10, the characteristic of log N is zero, so that log N is the 
same as its mantissa. Hence, a four-place table of mantissas is a table of the 
actual logarithms of all numbers (with at most three significant digits) from 1.00 
to 9.99. 

* The remainder of this book is arranged so that either four-place or fiv e-place 
computation may be omitted in whole or in part in the routine problems. In the tables 
of this book, auxiliary columns of proportional parts are given in the four-place as well 
as in the five-place tables, so that the details of interpolation in them are identical. 


28 


TRIGONOMETRY 


25. Five-place logarithms. In Table VIII we find the mantissa of 
log N correct to five decimal places if N is any number with at most 
four significant digits. 

Example 1. Find log .03162 from Table VIII. 

Solution. 1. To find the mantissa. On page 22 we find the first three digits, 
316, in the column headed “N.” In the row of 316, in the column headed by 2 
(the fourth digit of 3162) we find 996, the last three digits of the mantissa; its 
first two digits are 49, which appear in the column headed by 0. The complete 
mantissa is .49996. 

2. By Theorem III, the characteristic is — 2. Hence, 

log .03162 = - 2 + .49996 = 8.49996 - 10. 

Example 2. Find log 31,680 from Table VIII. 

Solution. 1. In the row of 316, in the column headed by 8 on page 22, we 
find *079; the asterisk “*” means that the first two digits of the mantissa are 
50, instead of 49 as at the beginning of the row. The mantissa is .50079. 

2. The characteristic is 4. Hence, log 31,680 = 4.50079. 

Example 3. Find N if log N = 5.40209 — 10. 

Solution. 1. The mantissa is .40209. We find the first two digits “40” in the 
column headed by Oon page 21. Among the entries belonging to this “40” we find 
209 in the row with 252 at the left margin and in the column headed by 4. Hence, 
the significant part of N is 2524. 

2. The characteristic of log N is (5 — 10), or — 5. Hence, by Theorem III, 
the first significant digit of N occurs in the fifth decimal place: N = .00002524. 

Comment. Example 3 could have been stated as follows: find the antilogarithm 
of (5.40209 - 10). 


EXERCISE 12 * 

Find the four-place logarithm of each number from Table V: 


1 . 

53.2. 

5. 

.00726. 

9. 

.000358. 

13. 

.00005. 

17. 

.0001. 

2. 

6.97. 

6. 

.0578. 

10. 

.00007. 

14. 

.47. 

18. 

.0089. 

3. 

.0163. 

7. 

8.45. 

11. 

6000. 

15. 

21,600. 

19. 

1100. 

4. 

.0279. 

8. 

.0035. 

12. 

9990. 

16. 

132,000. 

20. 

.3150. 


Find the antilogarithm of each four-place logarithm by use of Table V: 


21. 

2.3838. 

25. 9.9232 - 

10. 

29. 3.1553. 

33. 

6.7774 - 10. 

22. 

0.6684. 

26. 7.9445 - 

10. 

30. 4.1461. 

34. 

8.2201 - 10. 

23. 

0.8785. 

27. 5.9004 - 

10. 

31. 6.9494. 

35. 

- 2.3979. 

24. 

3.7938. 

28. 8.8129 - 10. 32. 7.9633. 36. 

* No interpolation is called for in this exercise. 

- 3.3010. 


LOGARITHMS 


29 


Find the five-place logarithm of each number: 


37. 198.7. 41. .01118. 

38. 18.56. 42. .2866. 

39. 1.389. 43. .2563. 

40. 2.633. 44. .0146. 


45. 59,600. 49. .801. 

46. 69,990. 50. 3.075. 

47. .00018. 51. 4168. 

48. .00009. 52. 10,070, 


53. 1,000,000, 

54. .000607. 

55. 10 - 69897 . 

56. 10 2 - 41326 . 


Find the antilogarithms of the following five-place logarithms: 


57. 1.25115. 

58. 2.47305. 

59. 4.68538. 

60. 3.77663. 


61. 9.42716 - 10. 

62. 8.58726 - 10. 

63. 7.49094 - 10. 

64. 9.09237 - 10. 


65. 0.66058. 

66. 5.83052. 

67. 3.61899. 

68. 0.48001. 


69. 6.55630 - 10. 

70. 5.68124 - 10. 

71. 0.11361. 

72. 0.30081. 


26. Interpolation in a table of mantissas is based on the assump¬ 
tion that, for small changes in N, the corresponding changes in log N 
are proportional to the changes in N. This principle of proportional 
parts for log N is merely an approximation to the truth but leads to 
results in interpolation which are sufficiently accurate for practical 
purposes. 

In finding a number N when log N is given, we agree to specify 
four or five significant digits in N according as we are using our four- 
place or our five-place table of mantissas. No greater refinement in 
the result is justified because the unavoidable error, which may arise, 
frequently will be as large as one unit in the last significant place in N, 
although rarely larger than one unit. 

The columns of proportional parts in the tables should be used to 
gain speed in interpolation , particularly in finding antilogarithms. 


27. Interpolation in a four-place table. 

Example 1. Find log 13.86 from Table V. 

Solution. By the principle of proportional parts, since 13.86 is ^ of the way 
from 13.80 to 13.90, hence log 13.86 is ^ of the way from log 13.80 to log 13.86. 


From table: log 13.80 = 1.1399. 1 Tabular difference is 

log 13.86 = ? [ 31 .1430 - .1399 = .0031. 

From table: log 13.90 = 1.1430. j .6(31) = 18.6, or 19. 

log 13.86 = log 13.80 + .6(.0031) *• 1.1399 + .0019 = 1.1418. 


Comment. We read .6(31) = 18.6 in the table of tenths of 31 in the column 
of proportional parts. We changed from 18.6 to 19 in the fourth decimal place 
because the table entries are accurate only to four decimal places. 



30 


TRIGONOMETRY 


Example 2. Find N if log N — 1.4709. 


Solution. 1. The mantissa is .4709, which is between the consecutive entries 
.4698 and .4713 in Table V; these mantissas correspond to 295 and 296. 

2. Since .4709 is H of the way from .4698 to .4713, we assume that the sig¬ 
nificant part of iV is xt °f the wa Y from 2950 to 2960. 


15 

[ U[ 




.4698, mantissa for 2950.“I 
.4709, mantissa for ? j 
.4713, mantissa for 2960. 


^ = .7, to nearest tenth. 
10 x = .7(10) = 7. 

2950 + 7 = 2957. 


3. Hence, the significant part of N is 2957. The characteristic of log N is 1. 
Hence, by Theorem II, N = 29.57. 

Comment. We obtained xt ~ -7 by inspecting the tenths of 15 in the column 
of proportional parts. We read 

.7(15) = 1C.5, or .7 = -8(15) = 12.0, or .8 = ~ 


Since 11 is nearer to 10.5 than to 12, hence H is nearer to .7 than to .8. 


28. Interpolation in a five-place table. 

Example 1. Find log 25.637 from Table VIII. 

Solution. By the principle of proportional parts, log 25.637 is ixy of the way 
from log 25.630 to log 25.640. 

From table: log 25.630 = 1.40875. ] Tabular difference is 

log 25.637 = ? > 17 .40892 - .40875 = .00017. 

From table: log 25.640 = 1.40892. j .7(17) = 11.9, or 12. 

log 25.637 = 1.40875 + .7(.00017) = 1.40875 + .00012 = 1.40887. 

Example 2. Find N from Table VIII if log N = 2.40971. 

Solution. 1. The mantissa is .40791, which is between the consecutive entries 
.40960 and .40976 in Table VIII; these mantissas correspond to 2568 and 2569. 

2. Since .40971 is xr of the way from .40960 to .40976, we assume that the 
significant part of N is H °f the way from 25680 to 25690. 



r.40960, mantissa for 25680.1 ~ 
|_.40971, mantissa for ? 


ii. = .7 ) to nearest tenth. 

16 

10 

x = .7(10) = 7. 


.40976, mantissa for 25690. 


25680 + 7 = 25687. 


3. Hence, the significant part of N is 25687. Since the characteristic of log N 
is 2, therefore N = 256.87. 

Comment. We obtained = .7 by inspection of the tenths of 16 in the 


column of proportional parts. We read 9.6 = .6(16), or .6 = and .7 — 
Since 11 is nearer to 11.2 than to 9.6, hence is nearer to .7 than to .6. 









LOGARITHMS 


31 


EXERCISE 13 


Find the four-place logarithm of each number from Table V: 


1. 2623. 6. 

2. 3676. 7. 

3. 6.968. 8. 

4. 9.892. 9. 

5. .1187. 10. 

26. 93,140,000. 


.001757. 

11. 

529.3 

.03865. 

12. 

15,910. 

.3073. 

13. 

.02555. 

131,500. 

14. 

.003158. 

4.023. 

15. 

18.680. 

27. 

.000000005673. 


16. 5322. 21. 43,190. 

17. 113.4 22. .00918. 

18. 1.298. 23. .000061. 

19. .9792. 24. 839,300. 

20. .06319. 25. 462,100. 

28. .000000000001431. 


Find the antilogarithm of each logarithm from Table V: 


29. 3.2367. 

30. 3.1395. 

31. 5.5511. 

32. 0.4228. 

33. 0.4906. 


34. 7.1247 

35. 6.3350 

36. 4.1436 

37. 9.6715 

38. 8.0255 


- 10. 39. 

- 10. 40. 

- 10. 41. 

- 10. 42. 

- 10. 43. 


2.9276. 

44. 

1.6016. 

45. 

4.5783. 

46. 

6.1640. 

47. 

0.0130. 

48. 


9.6270 - 10. 
5.9885 - 10. 
8.9935 - 10. 
7.1952 - 20. 
8.3358 - 20. 


Find the five-place logarithm of each number: 

49. 18,563. 52. 21.285. 55. 4.7178. 58. .89316. 61. 61.597. 

50. 25,632. 53. .30129. 56. 31.648. 59. .75362. 62. .071384. 

51. 5.3217. 54. .042087. 57. .073563. 60. 53.193. 63. 896,910. 

64. .0040063. 66. .00078651. 68. 966,910. 70. .000000000061. 

65. .0062873. 67. 1,300,600. 69. .00041569. 71. 5,000,600,000. 


Find the antilogarithm of each five-place logarithm: 


72. 2.21388. 

73. 3.21631. 

74. 1.33740. 

75. 2.05297. 


76. 9.65328 - 10. 

77. 8.12277 - 10. 

78. 7.94014 - 10. 

79. 9.77817 - 10. 


80. 6.03271. 

81. 5.45698. 

82. 0.97035. 

83. 0.28779. 


84. 9.00858 - 10. 

85. 3.33412 - 10. 

86. 6.24049 - 20. 

87. 8.73168 - 20. 


29. Computation of products and quotients. Unless otherwise 
specified, we shall assume that the data of any given problem are exact . 
Under this assumption, the accuracy of a product , quotient, or power 
computed by use of logarithms depends on the number of places in 
the table being used. The result is frequently subject to an unavoid¬ 
able error which usually is at most a few units in the last significant 
place given by interpolation. Hence, usually we must compute with 
at least five-place logarithms to obtain four-place accuracy, and with 
at least four-place logarithms to obtain three-place accuracy. Except 
when otherwise directed, in stating the solution of a problem we shall 
give all digits obtainable by interpolation in the table specified for use. 


32 


TRIGONOMETRY 


Example 1. Compute:* (.0631)(7.208)(.5127). 

Solution. 1. Let P represent the product. By Property I, page 23, we obtain 
log P by adding the logarithms of the factors. We obtain the logarithms of the 
factors from Table V, and then finally obtain P from Table V. 

( log .0631 = 8.8000 - 10 
( + ) log 7.208 = 0.8578 

' log .5127 = 9.7099 - 10 

log P = 19.3677 - 20 = 9.3677 - 10. 

2. Hence, P = .2332 [= antilog (9.3677 - 10)]. (From Table V) 


Example 2. 


Compute q — 


4.803 X 269.9 X 1.636 
7880 X 253.6 


Solution. First make a computing form, as given in heavy black type. By 
Property II, log q = ( log of numerator) — (log of denominator). 

( log 4.803 = 0.6815 j log 7880 = 3.8965 

(+) log 269.9 = 2.4312 \ log 253.6 = 2.4041 

I log 1.636 = 0.2138 _ log denom. = 6.3006. 

lognumer. = 3.3265 = 13.3265 — 10 
( —) log denom. - 6.3006 = 6.3006 

log q = ? = 7.0259 - 10. Hence, q = .001061. 

Comment. We foresaw that log q would be negative. To obtain log q in the 
standard form, we added 10 to 3.3265 and then subtracted 10. When it is necessary 
to subtract one logarithm from a smaller one, increase the characteristic of the smaller 
one by 10 and then subtract 10, to compensate. 

Example 3. Compute the reciprocal of 189. 

Solution. 1. Let R be the reciprocal. Then, R 

2. Recall that the logarithm of 1 to any base is 0. 

log 1 = 0.0000 = 10.0000 - 10 
(-) log 189 = 2.2765 = 2.2765 

log R = 7.7235 - 10. 


1 

189' 


Hence, R = .005290. 


★30. Cologarithms. The cologarithm of N, written colog N, is the 

logarithm of 1/ N. Since log 1 = 0, 

colog N — log -^ = 0 — log N. (1) 


Illustration 1. 


Colog .031 = log 


1 

.031 : 


log 1 = 10.0000 - 10 
(-) log .031 = 8.4914 - 10 
colog .031 = 1.5086. 


* Illustrative examples will be solved with the aid of four-place logarithms, but 
answers to exercises are given for both four-place and five-place computation. 









LOGARITHMS 


33 


To divide by a given number N is the same as to multiply by 1 JN. 
Hence, in computing a quotient we may add the cologarithm of each 
factor of the denominator to the logarithm of the numerator. Some 
computers prefer to use cologarithms in this way when possible, par¬ 
ticularly in certain trigonometric calculations. 


Illustration 2. To compute q 


16.08 

47 X .0158' 


q = 16.08 • ^ 


1 

.0158' 


log 16.08 = 1.2063 1 

log 47 = 1.6721; hence, colog 47 = 8.3279 - 10 [ (+) 

log .0158 = 8.1987 - 10; hence, colog .0158 = 1.8013 

log q = 1.3355. q = 21.65. 


EXERCISE 14 


Compute by use of four-place or five-place logarithms: 


1 . 

2 . 

7. 

12 . 

16. 


31.57 X .789. 3. .8475 X .0937. 

925.6 X .137. 4.* .0179 X .35641. 

675 „ 568.5 ^ 728.72 


8 . 


13.21 
.0421 
.53908' 
16.083(256) 
47(.0158) 


23.14 

13. 


9. 


10 . 


1 


1 


325.932 

17. 


5. * 925.618 X .000217. 

6. * 3.41379 X .0142. 


895 

14 ‘ 100,935 
9.32(531) 
.8319(.5685)' 


753.166 
9273.8 ' 

15. 
18. 


11 . 


.0894 


.6358 
.42173(.217) 
.3852(.956) ' 

5.4171(.429) 

18.1167(37)' 


Compute the reciprocal of the quantity: 

19. 853. 21. 985.3. 23. .00356278. 

20. 6498. 22. 131,573. 24. .00001399. 


25. .53819(.0673). 

26. .00073(.965). 


Compute the product or quotient: 

27. (- 84.75) (.00368) (.02458). 28. (- 16.8) (136.943) (.00038). 

Hint. Recall that only positive numbers have real logarithms. Compute as 
if all factors were positive and then determine the proper sign for the result by 
inspecting the signs of the factors. 


29. 

.038(— .29) (- .0065) 

30. 

2.71(— 1006.332) 

31. 

(a) Compute (853) (678). 


32. 

(a) Compute (.065) (.085). 

(b) 

33. 

(a) Compute .385 -4- 532. 

(&) 


5.6(— 3.9078)(- .00031) 

132(— 1.93) 

Compute (log 853) (log 678). 
Compute (log .065) (log .085). 
Compute (log .385) -4- (log 532). 


* Before finding the logarithm of a number N, we round off N to four or to five sig¬ 
nificant digits according as we are using four -place or five-place logarithms. This 
rule is followed because usually the tables do not justify greater refinement. Thus, 
for four -place computation, log .35641 = log .3564. For five -place computation, 
log 925.618 = log 925.62. 



















34 


TRIGONOMETRY 


SUPPLEMENTARY PROBLEMS 

Note. Any number N can be written in the form N = P • 10*, where 1 ^ P < 10 
and k is an integer, and where P has the same significant digits as N. In applied 
mathematics, the form N = P • 10* is used in writing very large or very small num¬ 
bers, or in clearly indicating how many digits are significant in a number N which 
is an approximate value of some quantity. Thus, if 153,000,000 is a number which 
has been explicitly determined only to five significant digits, we would write 
1.5300(10®) as the result; if accurate only to four places, we would write 1.530(10 8 ). 

Write in the form P • 10*, assuming that the number has been determined by 
experiment or computation to five significant digits: 

34. 3,037,200. 35. 15,600,000. 36. .0014326. 37. .056200. 

Write in ordinary form, and find its four-place logarithm: 

38. 10 9 (3.150). 39. 10 6 (2.165). 40. 10- 14 (3.146). 41. 10~ 15 (8.960). 

42. Suppose that N has been written in the form N = P • 10* described 
in the preceding note. Without using Theorems I, II, and III of page 25, 
prove that log P is the mantissa of log N and k is the characteristic. 

31. Computation of powers and roots. 

Example 1. Compute (.3156) 4 . 

Solution. By Property III, Section 19, log N 4 = 4 log N. 

log (.3156) 4 = 4 log (.3156) = 4(9.4991 - 10). 
log (.3156) 4 = 37.9964 - 40 = 7.9964 - 10. 

Therefore, (.3156) 4 = .009918. 

Example 2. Compute (a) 08351; ( b ) v^08351. 

Solution, (a). By equation 4, page 23, log v'iV = £ log N. 

log ^yS835l = ?°g 08351 = 8 9217 - 10 . 

6 6 

log v 7 . 08351 = 1.4870 - 1.6667 = - .1797. 

This result, — .1797, is inconvenient because it is not in the standard form for a 
negative logarithm. In order that the result after division by 6 may be in the 
standard form, we change (8.9217 - 10) by subtracting 50 from (- 10) to make 
it (— 60), and by adding 50 to 8.9217, to compensate for the subtraction. Then 

log = 8 9217 - 10 - S M 217 - 99 ,9.8203 - 10 . 

b 6 

Therefore, xC08351 = .6612. 

Solution. (6). log vC08351 = ^ - ^ 8351 = 8 9217 - 10. 

log v 7 .08351 = 28 ~ 921 J ~ 30 = 9.6406 - 10. 

Hence, ^.08351 = .4371. 



















LOGARITHMS 


35 


Comment. Remember the device just used: before dividing a negative logarithm 
by a positive integer, write the logarithm in such a way that the negative part, when 
divided by the divisor, gives — 10 as the quotient. 

Example 3. Compute q — (-^P^=L=Y 


,65.3V 146/ 

Solution. 1. Let ^represent the fraction: then? = inland log? — flogf* 7 . 
2. Notice that log (.5831) 3 = 3 log .5831; log V 146 = % log 146. 


log .5831 = 9.7658 - 10 
log 146 = 2.1644 
3 log .5831 = 9.2974 - 10 
( —) logdenom. = 2.8971 

log F = 6.4003 - 10; 


(+) 


log 66.3 = 1.8149 
i log 146 = 1.0822 


log q = 


2 log F 42.8006 - 50 


log denom. = 2.8971. 

2 log F = 2.8006 - 10 = 42.8006 - 50. 
= 8.5601 - 10. Hence, q = .03632. 


EXERCISE 15 

Compute by use of four-place or five-place logarithms: 

1. (17.5) 3 . 4. (3.1279) 4 . 7. Vl09. 10. V34168. 13. V.03107. 

2. (.837) 5 . 5. (1.9572) 6 . 8. V2795. 11. Vi857. 14. V.0001. 

3. (.0315) 3 . 6. (.0715) 3 . 9. V861. 12. V.0797. 15. V .00001. 

16. (.0138273)^. 18. V- 1.03. 20. (- 257)*. 22. (.357) i 

17. (700,928)i 19. V- 1796. 21. (34.168)i 23. (143.54)*. 


24. 5.713(2.56) 3 . 

25. .0956(13.2168) * 2 . 
56.3(4.317) 


30. 


31. 


32. 


21.4V521.923 

.0198 

(3.82616) 2 ’ 

758.32 


33 


27. 73.14V.98136. 

* Vi 


28. (14.6) 2 V.085. 

29. (3.456) 3 V.7316. 


89.1 


163 X .62 


34. yf- 


47.5317 


(46.3) 3 
39. (139)- 2 . 


35. 


031 X .964 
(25.73) (152) 3 


1893.32 
40. (1.03)- 5 . 41. (.98) -2 . 


36. 


37. 


38. 


»/(- 316)(.198) 


.756392 


V- 463.19 


V 16.3144 


54.2VT89 \ 2 


.157386 / 

42. (.067)" 3 . 43. (.0057)i 


44. 


45. 


(54.1)(1.03) 6 
47 56.3(10 2 - 31 ) 


V956.314 
10 3 66 X .0138' 


46. 


47. 


10-- 36 V.78 
(.983174) 2 ' 

10 -142 Vi387 
(57)(8.64) 2 ' 


48. 


49. 


/ (16.7)(.58) 2 \ 8 
V 65.131 J 


5731.84 y 


14.2Vi896/ 







































36 


TRIGONOMETRY 


MISCELLANEOUS PROBLEMS 

Definition. The geometric mean of n numbers is the nth root of their product. 
Thus, the geometric mean of M, N, and R is v 7 MNR. 

In each problem, find the geometric mean of the given numbers: 

50. 138; 395; 426; 537; 612. 52. .00138; .19276; .08356; .0131. 

51. 367; 157; 321; 598. 53. .857; .732; .629; .218; .345. 

Note. The formula A = P(1 + r)k gives the amount A due at the end of k 
years if a principal P is invested at the rate r (expressed as a decimal) compounded 
annually. If k is large, the computation of A by use of logarithms is very in¬ 
accurate unless log (1 + r) is given to many more decimal places than are desired 
in the final logarithm of A. To avoid such inaccuracy, some extremely accurate 
logarithms are given in Table XVI in the book of tables. 

54. Compute (1.04) 10 °, (a) entirely by use of four-place logarithms; 
(b) by using log 1.04 from Table XVI and then the four-place table. 


Compute the amount A in A = P(1 + r) k , for the given data, first using 

Table XVI for log (1 + r) and thereafter using Table V: 

55. P = $132; k = 30; r = .04. 57. P = $200; k = 40; r = .05. 

56. P = $150; k = 50; r = .03. 58. P = $125; k = 25; r = .06. 


59. The power P, in foot-pounds per second, available in a jet of water with 
a cross-section area of A square feet, discharging with a velocity of v feet per 
second, is P = \_{wv' i A) -f- 2g~\, where g = 32.16, and where w is the weight in 
pounds of one cubic fool of the water. Find the power of a jet whose cross- 
section area is 16.58 square inches if v = 5.76 and w = 62.23. 

60. If a body has fallen s feet from rest in a vacuum near the earth’s surface, 
the velocity v of the body in feet per second is given by v 2 = 2gs where g = 32.16, 
approximately, (a) Find the distance through which a body has fallen if 
v = 488. (6) Find the velocity of a body which has fallen 1740 feet. 

61. The time t in seconds for one oscillation of a simple pendulum whose 

length is l centimeters, is given by t = where g = 980. (a) Find t for 

0 

a pendulum for which l = 81.56 centimeters. (6) Find the length of a 
pendulum for which t = 3.75 seconds. 

62. Find [antilog (- 1.7328)] 2 . 

63. Let d be the diameter in inches of a short solid circular steel shaft which 
is designed to safely transmit H horse power when revolving at R revolutions per 

zl38H 

minute. A safe value for d is d = -y — ^ — Find the horse power which can 
be safely transmitted at 1150 revolutions per minute if d = 1.9834. 



LOGARITHMS 


37 


SUPPLEMENTARY PROBLEMS 

Compute by use of four-place or five-place logarithms: 


64. 

log 75 + 3 

„„ log 889 - 20 

68. 

y /45 - 364.1 

log 63.6 

66 ' 654 

(.9873) 3 + 16.3 

65. 

753 (log 8) 
log .189 

V23 - V 134.91 
b7 ' 453 X .110173 

69. 

567.89 + ^89 
57 + V.19538 

70. 

(3.57) 1 - 6 . 

71. (71.38) - 85 . 72. 6- 73 . 


73. (57.2)-- 32 . 

Hint for Problem 73. 

- .32 log 57.2 = - .5624 = 

9.4376 

- 10. 


74. (1.75) -3 - 14 . 75. (.0895)- 36 . 76. (.065)- 532 . 77. (.007)- 73 . 

78. An approximate formula for the maximum horse power P of the boiler 
which can be served by a chimney F feet high, whose cross-section area is A 
square feet, is P = 3.33(A — .6 Va)Vf. Find P for a chimney which is 
175 feet high and whose cross-section area is 9.65 square feet. 

79. An approximate formula for the volume V (cubic feel) of one pound of 
superheated steam which has a pressure of P pounds per square inch and an 

T 3 

absolute * temperature of T° Fahrenheit is V = .6490 -p — 22.58 P~±. Find 
V if P — 140 pounds and T — 900°. 

80. If a volume Vi of air at a pressure pi is compressed to a volume v 2 at a 
pressure p 2 , without losing any of the heat generated by the compression, then 

f v , \ 1 - 41 

p 2 = pJAJ . Under this condition, find the pressure necessary to com¬ 
press 117 cubic feet of air at a pressure of 25.8 pounds per square inch to 26 
cubic feet. , 

81. The weight w, in pounds of steam per second, which will flow through 
a hole whose cross-section area is A square inches, if the steam approaches 
the hole under a pressure of P pounds per square inch, is approximately 
w = .0165AP- 97 . (It is assumed that the steam flows into a reservoir where 
the pressure is -A f P.) How much steam at a pressure of 83.85 pounds per 
square inch will flow through a hole 12.369 inches in diameter? 

82. The growth of population in the United States is represented approxi¬ 
mately by the following formula ,t where P is the population in millions and t 
is the time in years, counting from 1780 a.d. 


P = 


197.27 

1 + 67.32e~- 0313< ' 


(e = 2.71828) 


Find P from this formula for (a) 1930 a.d.; (b) 2000 a.d. 

* Measured from absolute zero, which is - 459.6° Fahrenheit, 
f See The Biology of Population Growth , page 13, by Raymond Pearl. 












38 


TRIGONOMETRY 


★32. Exponential and logarithmic equations. A logarithmic equa¬ 
tion is one in which there appears the logarithm of some expression 
involving the unknown quantity. 

2^ 

Example 1. Solve for x : log x + log -r- = 6. 

o 

Solution. By use of Properties I and II of page 23, 
log x + (log 2 + log x — log 5) = 6. 

2 log x = 6 + log 5 — log 2 = 6.3980. (Using Table V) 

log x = 3.1990; x = antilog 3.1990 = 1581. (Using Table V) 


An equation where the unknown quantity appears in an exponent 
is called an exponential equation. Sometimes, an exponential equa¬ 
tion can be solved by taking the logarithm of each member and 
equating results. 


Example 2. Solve 13 2 *+ 2 = (356) (5*). 


Solution. 1. Log 13 2 * +2 = (2a; + 2) log 13; log 5 X = x log 5. 
2. Equate the logarithms of the two sides of the given equation: 


(2x -(- 2) log 13 = log 356 + x log 5; 
(2x + 2) (1.1139) = 2.5514 + .6990x; 


2.2278a; + 2.2278 = 2.5514 + .6990a;. 
1.5288a; = .3236. 

.3236 


1.5288 


- .2116. 


(Using Table II) 


log .3236 = 9.5100 - 10 
(-) log 1.529 = 0.1844 

log x = 9.3256 - 10 


Solve for x: 


★EXERCISE 16 


1 . 

2 . 

3. 

4. 

13. 


15. 


12* = 28. 5. 


51* = 569. 

5 2 * = 28(2*). 
15 3 * = 85(3*). 

log X 2 - log y 

(1.05)* - 1 
.05 


6 . 

7. 

8 . 

= 8.43. 
6.3282. 


.67* = 8. 
.093* = 12 
27* s = 54. 


9. (1.03)-* = .587. 

10. (1.04)-* = .642. 

11 . 6* a ~ 2 * = 18. 


6.52* 2 = 38.683. 


12. 5* 2+4 * = 17.64. 


14. log 5a; 2 + log - = 5.673. 


16. 


(1.03)* - 1 
.03 


= 12.675. 


Recall that A = P(1 + r) k is the equation of compound interest. 

17. When will the amount be $632.51, if $150 is invested today at the 
rate .05, compounded annually? 

18. When will the amount be $6500, if $2700 is invested today at the 
rate .025, compounded annually? 






LOGARITHMS 


39 


19. The vapor pressure, p , of liquid arsenic trioxide is given * by the 

2722 

equation log p =-+ 6.513, where T° Centigrade is the absolute 

temperature (absolute zero is — 273° Centigrade) and p is measured in 
millimeters of mercury, (a) What is the pressure when T = 500°? (6) At 
what temperature is the pressure 120 millimeters? 

20. The vapor pressure, p, in millimeters of mercury, of carbon tetra¬ 
chloride is well represented * by the equation 


log p = 


1706.4 

T 


+ 7.7760, 


where T° Centigrade is the absolute temperature, (a) At what tempera¬ 
ture does carbon tetrachloride boil, that is, when does p — 760 millimeters, 
the same as atmospheric pressure? (6) What is the pressure when the ab¬ 
solute temperature is 393°? 

21. A given radioactive substance decomposes at such a rate that, if 
B is the initial number of atoms of the substance and N is the number re¬ 
maining at the end of t hours, then N = Be~ kt , where A; is a constant and 
e = 2.71828 • • •. Given that, out of 21,000 atoms, 14,500 remain at the 
end of \ hour, (a) find k; (b ) find when only \ of the atoms will remain. 

22. The intensity, /, of a beam of light after passing through t centi¬ 
meters of a liquid which absorbs light, is given by I = Ae~ kt , where A is 
the intensity of the light when it enters the liquid, e — 2.71828 • • •, and k 
is a constant for any given liquid. Given that k = .12526, (a) how many 
centimeters of the liquid are sufficient to reduce the intensity of a beam to 
j of its original value? (6) How many centimeters are sufficient to absorb 
10% of the intensity? 


★33. Logarithms to bases different from 10. The equations 
x = logo N and a x = N are equivalent. Hence, if N and a are given, 
we can find logo N by solving the exponential equation N = a x for 
x by use of common logarithms. 

Note 1. Recall that the base of the natural system of logarithms is 
e = 2.718281828- • -. The following logarithms will be useful: 

logio e = 0.43429448. logio .43429448 = 9.63778431 - 10. 

Example 1. Find log e 35. 

Solution. Let x = log e 35; then, 35 = e x . On taking the common logarithm 
of both sides we obtain x logio e = logio 35. 

login 35 1.5441. log 1.544 = 10.1886 - 10 

X ~ logio e ~ 0.4343’ (-) log .4343 = 9.6378 - 10 

x = 3.555 = log e 35. log x = 0.5508. 

* See Mathematical Preparation for Physical Chemistry, page 68, by Farrington 
Daniels. 







40 


TRIGONOMETRY 


Theorem. If a and b are any two bases, then 

log a N = (loga fr)(log& N ). (1) 

Proof. Let y = log& N ; then N = b y . 

Hence, log a N = log a b y = y log a b = (log a b) (logs N). 

The number log a b is called the modulus of the system of base a 
with respect to the system of base b. Given a table of logarithms to 
the base b, we could form a table of logarithms to the base a by 
multiplying each entry of the given table by the modulus, logo b. 

Note 2. Material relating to logarithmic scales and the principle involved 
in slide rules is found in the Appendix, Note 1. 

★EXERCISE 17 

Find each logarithm, as in the preceding Example 1: 

1. loge 75. 3. logiB 33. 5. log e 10. 7. logs .097. 9. log. 8 23.8. 

2. loge 1360. 4. log 12 100. 6. logs 1.05. 8. log e .001. 10. log. 6 185. 

11. Find the natural logarithm * of (a) 4368.1; ( b ) 4.3681. 

12. Find the natural logarithm * of (a) 13,465; (6) 13.465. 

Find the modulus of each system with respect to the other: 

13. The Briggs system and the natural system of logarithms. 

14. The Briggs system and the system to the base 7. 

15. Prove that log a b = - — State this result in words. 

logs a 


★34. Graphs of logarithmic and exponential functions. Since the 


equations y = log a x and x = a v are 
graph. 

Illustration 1. Figure 16 is a graph of 
y = logo x when a = 2.71828- • -, but the 
general characteristics of the graph would 
not be different if a had any other value, 
greater than 1. This graph assists one in 
remembering the following facts: 

I. If 0 < x < 1, log a x is negative. 

II. If a has any value, log a 1=0. 

III. If x increases without limit, log a x 
increases without limit; if x approaches zero 


equivalent, they have the same 



Fig. 16. y = log e x 


, log a x decreases without limit. 


* If two numbers differ only in the position of the decimal point, their natural 
logarithms do not differ by an integer. This fact, as compared with the convenient 
properties of the characteristic and the mantissa of logio N, makes us prefer common 
logarithms for the purpose of computation. 














LOGARITHMS 


41 


★EXERCISE 18 

1. Graph y = logio x for 0 < x ^ 30. 

2. (a) Graph y = 2 X from x = — 5 to x = 5. ( b) From the graph, 

read the value of log 2 5; log 2 10; log 2 .5. 

Graph from x = — 5 to a; = + 5, using many values of x near to x = 0: 

3. y = 10 - * 2 . 4. y = 10“5. y = e~t. 6. y = 2~ x \ 

7. (a) Draw a graph of y = log. B * by graphing * = .5». (6) State 

facts similar to I, II, and III about logo x in case a < 1. 

REVIEW EXERCISE 19 

1. Define the logarithm of a number N to the base 10. 

Find the common logarithm of N in each problem: 

2. N = 10 3 - 67 . 3. N = 10 1 - 568 . 4. N = IQ- 2 - 46 . 5. N = lO" 1 - 503 . 


Find the number N whose logarithm is given: 


6. 

logio N = 4. 

10. 

log 3 N = 2. 

14. 

log 9 N = h 

7. 

logio N = 3. 

11. 

log 4 N = 3. 

15. 

log 25 N = b 

8. 

logio N = — 3. 

12. 

logs N = — 2. 

16. 

logs N = f. 

9. 

logio N = - 2. 

13. 

logs N = - 3. 

17. 

log 27 N = b 


Rewrite each equation in an equivalent logarithmic form: 

18. N = 3 4 . 19. N = 2®. 20. N = 5~ 3 . 21. N = lO" 7 . 

Find the logarithm of each number by applying the definition of a logarithm: 

22. 100. 23. 10,000. 24. 1000. 25. .1. 26. .001. 

Each number is the common logarithm of a number N. Only tell the charac¬ 
teristic and the mantissa of the logarithm: 

27. 5.68. 28. 4.32. 29. - 2.035. 30. - 3.06. 31. - .66. 

Tell the characteristic of log N if N is between the given numbers: 

32. Between 100 and 1000. 35. Between .1 and .01. 

33. Between 10,000 and 100,000. 36. Between .001 and .0001. 

34. Between 1 and 10. 37. Between 1 and .1. 

Compute by use of four-place or five-place logarithms: 

38 38 : 5 — ■ 39 (~ 35.68)(2.67) t 4( 0 V983.6. 41. ^.6421. 

624,132 .1438 

42. (1.03) 50 . 43. (1.04) -5 . 44. 157 log 46. 45. (log 895)/178. 

46. By use of the method of Section 19, page 23, prove that 
lo go \_M + (PQ)] = log M - log P - log Q. 






CHAPTER III 

LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 
35. Logarithms of trigonometric functions. 

Example 1. Compute 156 sin 21° 10' by four-place logarithms. 

Solution. In Table VI we find log sin 21° 10' (the logarithm of sin 21° 10'). 

log 156 = 2.1931 (From Table V) 

(+) log sin 21° 10' = 9.5576 - 10 (From Table VI) 

log 156 sin 21° 10' = 1.7507. Hence, 156 sin 21° 10' = 56.32. (From Table V) 

Comment. 1. The sine or cosine of any angle between 0° and 90°, or the tangent 
of any angle between 0° and 45°, or the cotangent of any angle between 45° and 90° is 
not zero and is less than 1. Therefore the logarithms of these functions have nega¬ 
tive characteristics. For abbreviation, the tables omit the “ — 10” which belongs 
with each of these logarithms which is tabulated. Hence, in using the columns 
labeled log sin, log cos, and log tan for 0° to 45°, we subtract 10 from each given 
entry. 

2. Table VI made it unnecessary to use Table VII in Example 1. 

To abbreviate the tables, we omit the logarithms of secants and 
cosecants. Hence, before computing a trigonometric expression by 
use of logarithms, if a secant (or cosecant) occurs as a factor, we change 
this function to the reciprocal of the cosine (or sine). 


36. Interpolation; four-place logarithms of trigonometric functions. 

Example 1. Find log cos 54° 46' from Table VI. 

Solution. We obtain log cos 54° 40' and log cos 54° 50' in Table VI. 


18 

' ["9.7622 «- log cos 54° 40H ~| 6' -J- 10' = .6. 

T ? <- log cos 54° 46' J b 10' x = .6(18) = 10.8. 

9.7604 <- log cos 54° 50' J 9.7622 - .0011 = 9.7611. 

Hence, log cos 54° 46' = 9.7611 — 10. 

Example 2. Find the acute angle a if log cot a = 9.8217 — 10. 

Solution. The positive part, 9.8217, is between the entries 9.8235 and 9.8208 
in the log coJ-column near 56°. Hence, a is between 56° 20' and 56° 30'. 

27 

1 „T9.8235 —> 56° 20H “j = .7, to nearest tenth. 

18 [_9.8217 -> a J 1 10' x = .7(10') = 7'. 

9.8208 -> 56° 30' J a = 56° 20' + 7' = 56° 27'. 


42 









LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 43 
37. Interpolation; five-place logarithms of trigonometric functions. 

Example 1. Find the five-place log cot 69° 32.3' by use of Table IX. 

Solution. 69° 32.3' is T 3 ^ of the way from 69° 32' to 69° 33'. The entries for 
69° 32' and 69° 33' are found on page 58 in Table IX. 


log cot 69° 32' -> 9.57197“] “ 


x = .3(39) = 11.7. 

log cot 69° 32.3' -> ? J * 

39 

9.57197 - .00012 = 9.57185. 

log cot 69° 33' -> 9.57158 


log cot 69° 32.3' = 9.57185 - 10. 


Comment. We subtracted .00012 because there is a decrease of .00039 in passing 
from 9.57197 to 9.57158. 


Example 2. Find the acute angle a if log cos a = 9.43254 — 10. 

Solution. The positive part, 9.43254, is between the entries 9.43278 and 9.43233 
in the log cos-column for 74° in Table IX. We see that a is ff of the way from 
74° 17' to 74° 18'. 


45 

’ 24 [ 




9.43278 

9.43254 

9.43233 


74° 17'" 
a. 

74° 18' 


ff = .5, to nearest tenth, 
x = .5(1') = .5'. 
a = 74° 17' + .5' = 74° 17.5' 


Note 1. Recall that tan a = —-— and hence 

cot a 

log tan a — log 1 — log cot a = 0 — log cot a = — log cot a. 

Therefore, any change in log tan a corresponds to a change of equal numerical 
value, but opposite sign, in log cot a. Hence, in Table IX, the single column headed 
“ cd” gives the common differences for the columns headed log tan and log cot. 

Note 2. Sometimes the values of trigonometric functions are referred to as 
natural functions, to emphasize the distinction between them and their logarithms. 
We could call Tables VII and XI tables of the natural trigonometric functions. 

38. The refinement of the results. In finding an angle a when the 
logarithm of some function of a is given, we agree to state the result 
to the nearest minute, or the nearest tenth of a minute, according as 
we are using the four-place Table VI or the five-place Table IX. 
When obtaining a logarithm from one of these tables, we agree to 
state the result to the same number of decimal places as are used in 
the tables. No greater refinement in our results is justified because 
the unavoidable errors, which may arise, frequently are as large as 
one unit in the last place we have agreed to specify, although usually 
no larger. Directions are given in the tables for avoiding interpo¬ 
lation where large errors might occur. 








44 


TRIGONOMETRY 


EXERCISE 20* 

1. (a) Find sin 12° 20' from Table VII. ( b ) Find the logarithm of the 
result of (a) by use of Table V. (c) Find log sin 12° 20' from Table I. 


Find the four-place logarithm of the function from Table VI: 


2. 

sin 5° 20'. 

7. 

cos 33° 54'. 

12. 

3. 

cot 55° 40'. 

8. 

sin 38° 23'. 

13. 

4. 

sin 8° 28'. 

9. 

cot 48° 55'. 

14. 

5. 

tan 32° 23'. 

10. 

sin 78° 37'. 

15. 

6. 

cot 41° 48'. 

11. 

cos 6° 52'. 

16. 


tan 35° 3'. 17. cos 45° 18'. 

cot 22° 7'. 18. cot 63° 39'. 

tan 37° 25'. 19. tan 11° 27'. 

sin 8° 23'. 20. cot 14° 22'. 

cos 21° 16'. 21. sin 53° 24'. 


Find the angle a by use of Table VI: 

22. log sin a = 9.0426 — 10. 

23. log tan a = 9.6946 — 10. 

24. log cos a = 9.4939 — 10. 

25. log cot a = 9.9595 — 10. 

26. log tan a = 9.7538 — 10. 

27. log sin a = 9.5665 — 10. 

28. log cot a = 0.0845. 

29. log cos a = 9.8893 — 10. 

30. log cos a = 9.4113 — 10. 

31. log cot a = 9.8752 — 10. 


32. log sin a = 9.9266 — 10. 

33. log tan a = 0.2007. 

34. log cot a — 0.1058. 

35. log cos a = 8.9570 — 10. 

36. log sin a — 9.7384 — 10. 

37. log tan a = 0.3141. 

38. log cos a = 9.6347 — 10. 

39. log tan a = 8.9899 — 10. 

40. log cos a — 8.9642 — 10. 

41. log sin a = 9.5470 — 10. 


Find the five-place logarithm of each function from Table IX: 

42. sin 23° 17'. 

43. cos 48° 19'. 

44. sin 17° 23.6' 


45. tan 27° 39.7'. 

46. cot 22° 33.3'. 

47. cos 44° 48.9'. 

48. tan 51° 43.5'. 


49. sin 68° 19.8'. 

50. cos 80° 39.6'. 

51. cot 56° 8.1'. 

52. sin 38° 23.7'. 

53. cos 62° 33.7'. 

54. tan 78° 3.2'. 

55. cot 61° 15.8'. 


56. cos 20° 4.1'. 

57. sin 80° 0.3'. 

58. tan 46° 0.7'. 

59. cot 38° 19.4'. 

60. sin 54° 31.5'. 

61. cos 14° 8.2'. 

62. cot 88° 43.4'. 


Find the acute angle a by use of the five-place Table IX: 


63. log tan a — 9.49474 — 10. 

64. log sin a = 9.04149 — 10. 

65. log cot a — 9.06445 — 10. 

66. log tan a = 0.86333. 

67. log tan a = 9.48750 — 10. 

68. log sin a — 9.38941 — 10. 

69. log cos a = 9.52551 — 10. 

70. log cot a = 9.72159 - 10. 

71. log cot a = 0.29239. 

72. log cos a = 9.97822 — 10. 

* The instructor may direct omission 


73. log sin ol — 9.39573 — 10. 

74. log tan a = 0.44796. 

75. log cot a = 0.97621. 

76. log cos a = 9.52714 — 10. 

77. log tan a = 9.59743 — 10. 

78. log sin a = 9.19516 — 10. 

79. log cot a — 9.55877 — 10. 

80. log cot a = 1.02389. 

81. log sin a = 9.97921 — 10. 

82. log cos ol = 8.99348 — 10. 

the four-place, or five-place, problems. 


LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 45 

39. Logarithmic solution of a right triangle. We recall the follow¬ 
ing formulas for a right triangle lettered as in Figure 17. 

(1) a 2 + b 2 = c 2 . (2) a + j8 = 90°. 

(3) tan a. = ~ = cot /3. (4) cot a = - = tan p. 

o a 

(5) sin a = - = cos /3. (6) cos a = - = sin 0. 

c c 

For our convenience when using logarithms, we alter (1) by writ¬ 
ing a 2 = c 2 — 6 2 and 6 2 = c 2 — a 2 ; then 

a 2 = (c — 6)(c + 6), or a = V(c — b) (c + 6); (7) 

b 2 = (c — a)(c + a), or & = V(c — a)(c + a). (8) 

Either (7) or (8) is convenient for checking the solution of a right 
triangle by logarithms, if we avoid using (7) or (8) in solving the triangle. 

Example 1. Solve triangle ABC if a = 135.1 and b = 367.2. 

Solution. We arrange the computing form, which is given in heavy type, 
before looking up any logarithms. 


B 



AbC 
Fig. 17 


Formulas 

Computation 


Data: a = 135.1; c = 

367.2. 


log a = 2.1306 

(Table V) 

a 

(— ) log c = 2.5649 

(Table V) 

sin a = -• 
c 

log sin a = 9.5657 — 10 



a = 21° 35'. 

(Table VI) 

b 

log c = 2.5649 

(Above) 

cos a = - > or 
c 

(+) log cos a = 9.9684 — 10 

(Table VI) 

b = c cos a. 

log b = 2.5333; b = 341.4. 

(Table V) 

P = 90° - a. 

P = 90° - 21° 35' = 68° 25'. 


S ummar y. a = 21° 35'; P = 68° 25'; b — 341.4. 


Check. a = V(c — b)(c + b ), or log a = | log [(c — 6)(c + &)]. 
c - b = 25.8 log (c - 6) = 1.4116 

c + 5 = 708.6 (+) log (c + 6) = 2,8504 

log [(c - &)(c + 6)] - 4.2620. 

log q = 2,1306 —» jlog [(c - &)(c + &)] = 2.1310. 

Comment. The check is satisfactory. The difference (2.1310 — 2.1306), or 
.0004, could result from an error of less than one unit in the fourth significant digit 
of b, because this would affect the third significant digit of (c — b). We cannot 
expect a refined check from (7) if b is nearly equal to c. 



















46 


TRIGONOMETRY 


Example 2. Solve triangle ABC if a = .8421 and a = 27° 40'. 
Solution. The student should check the following solution: 


Formulas 

Computation 


Data: a = .8421; a = 27° 40'. 

0 = 90° - oc. 

0 = 90° - 27° 40' = 62° 20'. 

b 

- = cot a, or 
a 

b = a cot a. 

log a = 9.9254 - 10 (Table Y) 

(+) log cot a = 0.2804 (Table VI) 

log b = 0.2058; b = 1.606. 

a 

- = sm a, or 
c 

_ a 
sin a 

log a = 9.9254 - 10 
(—) log sin a = 9.6668 — 10 

log c = 0.2586; c = 1.814. 

Summary. 0 = 62° 20'; b = 1.606; c = 1.814. 


EXERCISE 21 


Solve triangle ABC of Figure 17 by use of four-place logarithms, and check: 


1. 

a = 

15.7; a = 36° 20'. 

11. 

a = 31° 7'; a 

= 1.415. 

2. 

c = 

.943; 0 = 62° 40'. 

12. 

a = 4.263; b 

= 8.377. 

3. 

a = 

.3590; b = .6611. 

13. 

c = 9.156; a 

= 4.113. 

4. 

c — 

3.632; a = 1.351. 

14. 

a = 23° 9'; c 

= 10.03. 

5. 

b = 

82.7; a = 31° 42'. 

15. 

0 = 49° 53'; a = 41.08. 

6. 

c = 

41.9; a = 14° 18'. 

16. 

c = .0314; b 

= .0195. 

7. 

c = 

.913; a = .267. 

17. 

a = .024614; 

a = 25° 19'. 

8. 

b = 

.573; a = .385. 

18. 

c = 41,318; a 

= 32,142. 

9. 

a = 

71° 26'; c = 4283. 

19. 

a = .57; b = 

10.03. 

10. 

0 = 

68° 42'; a — 1935. 

20. 

a = 1.36; c — 

: 21.85. 

Solve triangle ABC of Figure 17 by 

use 

of five-place logarithms, and check: 

21. 

a = 

23.18; 0 = 47° 17'. 

31. 

a = 31° 24.7'; 

a = 1.6315. 

22. 

c = 

.685; a = 29° 43'. 

32. 

a = 2.1523; b 

= 4.1392. 

23. 

c — 

.43580; a = .29971. 

33. 

c = 915.62; a 

= 411.37. 

24. 

a = 

40,986; b = 43,216. 

34. 

a = 68° 39.2'; 

c = 1000.3. 

25. 

c = 

41.952; a = 25° 37'. 

35. 

0 = 38° 47.9'; 

c = 2007.1. 

26. 

b = 

88.783; 0 = 49° 8'. 

36. 

a = 53° 22.6'; 

b = 93.142. 

27. 

a = 

.5731; b = .6298. 

37. 

a = .94283; a 

: = 29° 14.5'. 

28. 

c = 

.3675; a = .1943. 

38. 

c = 831.423; 

a = 613.578. 

29. 

a = 

25° 8'; c = 37.857. 

39. 

a = 2.17; b = 

103.4. 

30. 

P = 

43° 17.6'; a = 42.93. 

40. 

a = 6.3; c = 

229.5. 












LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 47 


SUPPLEMENTARY PROBLEMS 


K 



Note. In Figure 18, triangle ABK is isosceles. 
The perpendicular KD to the base divides triangle 
ABK into two equal right triangles. In triangle ADK, 
x = \k and Z AKD — §y. 

Find the unknown sides or angles of triangle 
ABK under the given conditions. Use four-place 
or five-place logarithms as directed by the instructor: 

41. y = 68° 20'; y = 456. 

Hint for Problem 41. sin 34° 10'= -• 

V 


42. a = 56° 40'; km 288. 

43. a = 48° 30'; y = 630. 46. y = 46° 56'; h m 250. 

44. y = 77° 40'; y = 3.188. 47. y = 49° 28'; h = 84. 

45. k = 32.68; y = 23.48. 48. k = .63284; y = .83172. 


Note. In any oblique triangle ABC, we let a, ft, and y be the angles at A, 
B, and C and let a, b, and c be the lengths of the sides opposite to vertices A, B, 
and C, respectively. 


Solve the oblique triangle ABC ; use four-place or five-place logarithms as 
directed by the instructor: 

[ b = 275, f a = 315.7, 

49. I a = 46° 26', 50. { a = 42° 32', 

1 y - 103° 54'. [ ft = 59° 48'. 

Hint for Problem 49. 1. See Figure 19. 

Drop a perpendicular from C to AB. Since 
a + ft + y = 180°, find ft. Fig. 19 

2. In triangle ADC, compute x and log h. 

3. In triangle CDB, compute y and o; then c = x + y. 



(c m .9438, 

(c = 68.452, 

f a = 14.56, 

la = 43° 18', 

53. J 6 = 31.267, 

55. 6 = 19.37, 

1 y = 61° 44'. 

[ a = 28° 15.6'. 

[ ft = 110° 50' 

f a = 221.8, 

[ a = 9.486, 

[c = 6.48, 

b = 434.7, 

54. lc = 3.147, 

56. 6 = 3.15, 

[ y = 47° 53'. 

[ ft = 129° 16'. 

1 y = 123° 20' 


40. Problems with approximate data. In numerical applications 
of mathematics, we usually deal with approximate data. The degree 
of accuracy of the data and the nature of a problem must be carefully 
considered in deciding what places are significant in the results. 

In data concerning triangles, it is roughly true that three-place, 







48 


TRIGONOMETRY 


four-place, and five-place accuracy in the lengths of sides correspond 
to accuracy to the nearest 10', 1 ', and .1', respectively, in the angles. 
In stating that a set of data has, for instance, four-place accuracy, we 
shall mean that the angular data are accurate to the nearest minute. 

Hereafter in this book, assume that the data in any applied problem, 
stated in words, are not exact but are the result of measurement with 
accuracy limited to the significant places which appear in the data. 

Note 1. The instructor may direct that each applied problem be solved by 
use of the least extensive tables which will give results as accurate as the data. To 
accomplish this, use four-place or five-place tables according as the data are accu¬ 
rate to three or to four places; then round off final results to the same number of 
places as were specified in the data. This procedure usually restricts any unavoid¬ 
able errors of computation to the extra places in the results which are eventually 
rounded off. Answers will be given as they appear after being rounded off 
to a number of significant places which appears reasonably justified by the ac¬ 
curacy of the data. 


EXERCISE 22* 

1. From the top row in a football stadium, 85 feet above the ground, 
the angle of depression of the center of the field is 32° 20'. Find the air-line 
distance from the top row to the center of the field. 

2. A railroad track runs due south and rises at an angle of 3° 38' from 
the horizontal. In traveling 18,640 yards along the rails, how far south and 
how far up does a train go? 

3. In flying upward for 1260 yards along a straight path, an airplane 
rises 156 yards. Find the angle of inclination of the path to the horizontal. 

4. Find the angle of elevation of the sun if a pole 90.7 feet tall casts a 


horizontal shadow 47.5 feet long. 

5. Find the pitch of a gable roof 45.7 feet b 

wide whose peak is 17.6 feet above the eaves; ./Tn. 

also, find the angle which a side of the roof makes j n. 

with the horizontal. ! \ 

Hint. In Figure 20, the pitch is defined as AB/DC. P a c 

6. If the pitch of a gable roof is .7361, find Fig. 20 

the angle which a side makes with the horizontal. 


7. If a gable roof is 38.9 feet wide and if its sides are inclined 48° 30' 
from the horizontal, how high is the peak above the eaves? 

8. A porch whose roof is 10.6 feet above the ground projects 15.3 feet 
from the wall of a house. Find the length of the shortest ladder which 
would reach over the porch to a window 55.6 feet above the ground. 


* In some problems it may be convenient to solve without logarithms. 



LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 49 


9. A regular octagon is inscribed 
in a circle whose radius is 36.8 feet. 

Find the length of a side of the octa¬ 
gon and the radius of its inscribed 
circle. 

Hint. 1 . In Figure 21, connect the 
center of the circle to the vertices, thus 
forming eight equal isosceles triangles. 

2. In triangle ABK, each of the 
equal sides is 36.8 feet long; the vertex 
angle is i(360°), or 45°, and the altitude 
AH is the radius of the inscribed circle. 

10. Each side of a regular hexa¬ 
gon (six sides) is 50.8 inches long. 

Find the radii of its inscribed and 
circumscribed circles. 

11. Each side of a regular decagon (ten sides) is 127 inches long. Find 
the radii of its inscribed and circumscribed circles. 

12. A regular pentagon (five sides) is circumscribed about a circle whose 
radius is 48.7 inches. Find the length of a side of the pentagon. 

13. If the inscribed circle in a regular polygon of seven sides has a radius 
of 103 inches, find the length of a side of the polygon. 

14. Find the area of a parallelogram the lengths of whose sides are 75.3 
feet and 40.7 feet, if one angle is 98° 36'. 



Fig. 21 


Without using any trigonometric tables, find the radii of the inscribed and 
circumscribed circles for the specified polygon: 

15. An equilateral triangle; each side is 18.6 inches long. 

16. A square; each side is 54.3 inches long. 

17. Find the area of a parallelogram the lengths of whose sides are 
150.6 feet and 235.3 feet, if one angle is 127° 26'. 

18. Find the angles of intersection of the diagonals of a rectangle whose' 
dimensions are 6.86 feet by 14.36 feet. 

19. A tower stands on top of a cliff 653 feet above a horizontal plane. 
From a point A in the plane, the angles of elevation of the top and the 
bottom of the tower are 47° 28' and 43° 36'. Find the height of the tower. 

20. A boat is anchored 6750 yards due east of a north-south coast line. 
From the boat, the direction lines to two points A and B on the beach are 
inclined, respectively, 45° 27' and 47° 33' north of a line pointing west. 
Find the distance between A and B. Use Tables XI and VIII. 









50 


TRIGONOMETRY 


SUPPLEMENTARY PROBLEMS * 

21. From two points 235.7 yards apart on a horizontal road running due 
east from a mountain, the angles of elevation of its top are, respectively, 
43° 27' and 30° 18'. How high above the road is the mountain top? 

Outline of solution. 1. In Figure 22 we desire h. We are given a, a, and (3. 
2. From triangle BCT, x = h cot /3. (1) 

T 3. From triangle ACT, cot a = or 

a + x = h cot a. (2) 

4. Solve (1) and (2) for h by substituting (1) in (2): 

(3) 



a + h cot /3 = h cot a; 
a = h cot a — h cot j8. 


h = 


cot a — cot j8 

22. From the top of a cliff above a horizontal plane we observe two 
points due north which are 376 feet apart. If the angles of depression of 
the points are 32° 10' and 26° 33', find the height of the cliff. 

23. An airplane is flying horizontally due east from us at a speed of 135 
miles per hour. We observe the angle of elevation of the airplane to be 
23° 17'; twelve seconds later its angle of elevation is 21° 39'. How high 
above us is the airplane flying? 

24. An airplane, which is flying horizontally due south from us, is 
towing an artillery kite-target 1000 feet behind it. At a certain moment, 
the angles of elevation of the target and the airplane are 43° 20' and 39° 56'. 
How high above us is the airplane flying, if the target and the airplane are 
at the same height? 

25. From the mast of a warship 108 feet above water level we observe 
points T and B at the top and the base of a cliff at the edge of the water; 
the angle of elevation of T is 37° 46' and the angle of depression of B is 
18° 27'. Find the height of the cliff and the distance from the warship to 
B, given that B and T are in the same vertical plane. 

26. In Figure 23, suppose that side a and 
angles j3 and y are given. If h is the length of the 
altitude from A to BC, prove that 


/1 \ 
c / 

\b 

y h\ 


/Ax \ D 

/tNv 


h = 


cot j3 + cot y 


Fig. 23 


27. A tower 20.7 feet high stands at the edge of the water on a bank 
of a river. From a point directly opposite on the other bank, above water 
level, we observe that the angle of elevation of the top of the tower is 27° 17' 
and the angle of depression of the image of its top in the water is 38° 12'. 
Find the width of the river. 


* Use Table XI if natural values of functions are needed. 








LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 


51 


★41. Projections. Let CD be a segment of a line in the same plane 
as a second line OL. Then, if A and B are the feet of the perpendicu¬ 
lars from C and D to OL, we call AB the projection of CD on OL. 
In Figure 24, let a be the acute angle between OL and the line of CD. 
Then, since AB = CH, from triangle CHD we obtain 

AB — CD • cos a; (1) 

or, the projection of a segment CD on the given line is equal to CD 
times the cosine of the acute angle between CD and the line. 

Let OM be perpendicular to OL. 
In Figure 24, the projection of CD 
on OM is EF and EF = HD ; hence 
EF = CD • sin a. (2) 

Illustration 1. If a train travels 3000 
feet along a straight grade inclined 2° 37' 
from the horizontal, then the lengths of 
the horizontal and vertical projections of 
the path of the train are 3000 cos 2° 37' 
and 3000 sin 2° 37', respectively. 

Let AB == x and EF = y. Then, CH - x and HD = y. There¬ 
fore, CD = Vr + y 2 , Thus, if the projections of CD on two per¬ 
pendicular lin es are known, we can determine the length of CD. 

★42. Bearing of a line. In a horizontal plane, let us represent any 
direction by an arrow or directed line segment radiating from some 
point 0. In the language of the surveyor, the direction of a directed 
line is called its bearing and is described by specifying the acute angle 
which the line makes with the north or the 
south arrow. 

Illustration 1. In Figure 25, the bearing of OA is 
20° west of north, abbreviated N 20° W. The bear¬ 
ing of OB is S 40° W. To describe the bearing of 
a line, we first write the letter N (or S), then the 
acute angle between the given line and ON (or 
OS), and finally write E or W to show on which 
side the given direction falls. 

Example 1. In Figure 25, find the bearing of 
K as seen from 0 if K is 30.6 miles north and 
78.3 miles east of 0. 

Solution. 1. OD = 78.3; DK = 30.6 = OF. 

2. From triangle FKO, we find a — 68° 39'. The bearing of OK is N 68° 39' E. 


N 



Fig. 25 










TRIGONOMETRY 


52 . 


★EXERCISE 23 

Find the horizontal and vertical projections of A B if it has the given length 
and inclination from the horizontal: 

1. 137.5 feet; inclination = 17° 18'. 3. .1638 feet; inclination = 49° 7'. 

2. 4326feet; inclination = 58°42'. 4. .04361 feet; inclination = 75° 8'. 

Find the distance between A and B and the inclination of A B to the horizon¬ 
tal, if the horizontal and vertical projections of AB are as stated: 

5. Horizontal = 17.8 feet; vertical = 23.6 feet. 

6. Horizontal = .8135 feet; vertical = .4216 feet. 

7. Horizontal = .06143 feet; vertical = .05813 feet. 

8. Horizontal = .01326 feet; vertical = .09345 feet. 

How far north or south and how far east or west is M from P? 

9. M is 385 miles from P in the direction N 43° 20' E. 

10. M is 16.2 miles from P in the direction S 27° 50' W. 

11. P is 23.7 miles from M in the direction S 49° 10' E. 

12. P is 896 miles from M in the direction N 73° 40' W. 

Find the distance MP and the bearing of M from P: 

13. M is 58.9 miles north and 32.7 miles east of P. 

14. M is .895 miles south and .623 miles west of P. 

15. P is .648 miles east and .572 miles south of M. 

16. P is 138 miles north and 289 miles east of M. 

Points M and P are in the same horizontal plane; P cannot be seen from M 
and MP cannot be measured directly. Find the distance and direction of P 
from M if a surveyor goes from M to P by the indicated path: * 

17. First 25.8 miles N 28° 10' E; then 43.7 miles N 47° 40' E. 

Hint. Draw a figure. First find the east-west and the north-south projections 
of each part of the path. Combine the east or west projections to find how far 
east or west P is from M. 

18. First 12.5 miles N 83° 30' E; then 65.8 miles S 48° 40' W. 

19. First 168 miles S 27° 50' E-, then 43 miles N 21° 30' E. 

20. First 43 miles S 75° 10' W; then 198 miles N 43° 20' E. 

21. First 578.3 feet S 25° 15' W) then 1321 feet N 63° 17' W; then’ 
735.6 feet S 29° 21' E. 

22. First 6349 feet N 73° 39' E; then 2137 feet S 62° 3' W; then 
5378 feet S 21° 17' W. 

* Additional problems and terminology relating to surveying are found in the Ap¬ 
pendix, Note 9. 


LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 


53 


N 


★43. Vectors. A vector is a quantity which possesses magnitude 
and direction. Hence, we can represent any 
vector by an arrow whose length is propor¬ 
tional to the magnitude and whose head 
points in the direction of the vector. w ~ 

Illustration 1. In Figure 26, OF represents a force 
of 50 pounds pulling on a body at 0 in the direction 
N 70° E. OD represents a force of 25 pounds acting ® 

<S 50° E. Fig. 26 



r F 

o 

D 


By definition, the sum, or the resultant, of two vectors, OF and OP, 
is the vector OR which is the diagonal of the 'parallelogram of which OF 
and OP are adjacent sides. We call OF and OP the components of 
OR along these lines. If OP and OF in Figure 27 represent forces 
r acting simultaneously on 0, then it is 
proved in physics that the effect of these 
forces is the same as that due to their resultant 
acting alone on 0. 

jr IG 27 Note 1. The preceding statement is referred 

to as the parallelogram law for the composition of 
forces. A similar law holds for the composition of velocities, or of various other 
vector quantities. 



The components of a given vector OR along two perpendicular 
lines OF and ON are the vectors obtained by N 
projecting OR on OE and ON. In Figure 28, 
the magnitudes of the components are 

OA = OR • cos a; OB = OR • sin a. 

The given vector OR can be thought of as the 
resultant of its components OA and OB. 



N 



Example 1. A body is acted upon simultaneously by 
the following forces in the given directions: 75 lb., east; 
125 lb., north. Find the magnitude and direction of 
their resultant. 

Solution. 1. Outline. In Figure 29, OR is the resultant. 

OD = BR = 75; OB = 125. 

Find (3 from tan /3 = fffa. 

Find OR from OB = OR cos @. 

2. Computation. The student should show that 
OR = 145.8; p = 30° 58'. 


Hence, the resultant is a force of 145.8 lb. acting N 30° 58' E. 











54 


TRIGONOMETRY 


★EXERCISE 24* 

Find the horizontal and vertical components of the force: 

1. 153.8 lb., acting upward at inclination of 33° 26' from horizontal. 

2. 2183 lb., acting downward at inclination of 53° 27' from horizontal. 

Find the north-south and east-west components of the force: 

3. 42.65 lb., acting N 31° 10' E. 5. 318.2 lb., acting S 21° 36' E. 

4. 5756 lb., acting S 47° 30' W. 6. 1329 lb., acting N 58° 23' W. 

A body is acted on simultaneously by the given forces in the specified direc¬ 
tions. Find the magnitude of the resultant and its direction, or its deviation 
from the horizontal: 

7. 17.8 lb., vertically upward; 32.6 lb., horizontally. 

8. 39.5 lb., vertically downward; 78.5 lb., horizontally. 

9. 162 lb., north; 53.7 lb., east. 11. 638 lb., south; 217 lb., west. 

10. 38.8 lb., north; 14.3 lb., west. 12. 528 lb., south; 133 lb., west. 

13. A boat is sailing east at the rate of 18 miles per hour. A man walks 
north across the deck at the rate of 3 miles per hour. Find the direction of 
his motion and his speed with respect to the surface of the earth. 

14. A river flows at the rate of 7.8 miles per hour. A man, who can row 
4 miles per hour in still water, rows directly across the current. Find the 
rate and the direction of his motion. 

Find the magnitude and the direction of the resultant of the given forces with 
the given directions acting simultaneously on a body: 

15. 278 lb., N 47° 20' E; 475 lb., N 68° 30' E. 

Hint. 1. Find the east-west and the north-south components of each force. 

2. Each component of the resultant is the sum of the corresponding com¬ 
ponents of the given forces. 

16. 57.3 lb., N 21° 10' E; 158 lb., N 49° 20' E. 

17. 125 lb., S 82° 40' E; 389 lb., 8 37° 50' E. 

18. 47 lb., N 23° 10' W; 163 lb., S 84° 30' W. 

19. 433 lb., N 35° 40' E; 119 lb., S 79° 50' W. 

20. 369 lb., N 80° 20' W; 187 lb., S 26° 30' E. 

21. 137 lb., N 17° 10' W; 426 lb., 8 85° 30' E. 

22. 157 lb., N 35° 20' E; 73 lb., N 23° 40' W; 238 lb., S 85° 50' E. 

23. 841 lb., S 18° 20' W; 348 lb., S 72° 30' E; 195 lb., N 38° 0' E. 

* In problems involving forces, we disregard friction and other complicating 
features. Unless otherwise specified, all forces considered are supposed to act 
horizontally. 


LOGARITHMIC SOLUTION OF RIGHT TRIANGLES 55 

24. Find the force which is exactly sufficient to keep a 1000-pound 
weight from sliding down a plane inclined 
47° to the horizontal. 

Hint. Gravity creates the force AW of 1000 
lbs., which is the resultant of its components AK 
and AH, acting along and perpendicular to the 
plane. (Force AH merely creates pressure against 
the plane which is counteracted by the supports 
of the plane.) AK tends to move the weight A 
downward. We must apply a force AF equal in 
magnitude to AK but in the opposite direction. 

25. A 150-pound shell for a battery of artillery is dragged up a runway 
inclined 42° to the horizontal. Find the pressure of the shell against the 
runway and the force required to drag the shell. 

26. A motor truck weighing 6875 pounds, loaded, moves up a bridge 
inclined 7° 32' from the horizontal. Find the pressure of the truck against 
the bridge. 

27. An automobile weighing 2600 pounds stands on a hill inclined 
25° 36' from the horizontal. How large a force must be counteracted by 
the brakes of the automobile to prevent it from rolling downhill? 

28. Cables running due north from a telegraph pole AP, in Figure 31, 
create a horizontal pull of 580 pounds at the top of the pole. A supporting 

cable PC is run due south to the ground. Find the tension 
in CP if CP is inclined 49° 50' from the horizontal. (The 
tension is a, force, pulling in the direction PC, whose hori¬ 
zontal component counteracts the pull on the pole and 
whose vertical component adds to the pressure of the pole 
against the ground.) 

29. A guy wire 78 feet long runs from the top of a tele¬ 
graph pole 56 feet high to the ground and pulls on the pole with a force of 
290 pounds, (a) What is the horizontal pull of the wire on the top of the 
pole? (6) What vertical force does the wire exert as an addition to the 
pressure of the pole against the ground? 

30. What force must be exerted to drag a 150-pound weight up a slope 
which inclines 25° from the horizontal? 

31. What is the largest weight which a man can drag up a slope which 
inclines 35° from the horizontal if he can pull with a force of 125 pounds? 

32. A man finds that his strength is just sufficient to drag a 200-pound 
weight up a certain slope. Find the angle at which the slope is inclined to 
the horizontal if the man is able to exert a pull of 140 pounds. 

33. A man wishes to raise a 300-pound weight to the top of a wall 15 
feet high by dragging the weight up an incline. What is the length of the 
shortest inclined plane he can use if his pulling strength is 135 pounds? 



c a 
Fig. 31 








CHAPTER IV 

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 

44. A number * scale. In Figure 32 all positive numbers are repre¬ 
sented in order by the points to the right of A and all negative num¬ 
bers by the points to the left of A. If M and N are two numbers 
such that M < N, this means geometrically that M is to the left of 
N in Figure 32. 

M A at 

I—I—I—I—HH—I—I—I—I—I—I—I—I—I—I—I—HH—I—I 

—7 —6 —5 —4 —3 —2 —1 0 1 2 3 4 6 6 

Fig. 32 

Illustration 1. 3 < 5; — 3 < 0; — 3 < 2; — 2 > — 5. 

To say that M < 0 means that M is negative. 

To say that N > 0 means that N is positive. 

45. Numerical value. By definition, the numerical value, or the 
absolute value, of a number R is R itself, if R is 'positive, and is the 
negative of R, if R is negative. That is, the numerical value of R is a 
positive number regardless of whether R is positive or negative. 

Illustration 1. The numerical value of 7 is 7, and of — 7 is also 7. 

We say that R and S are numerically equal if they have the same 
numerical value; then, R and S do not differ except perhaps in sign. 

Illustration 2. The numbers 3 and — 3 are numerically equal. 

46. Directed lines. Suppose that, on a certain straight line, we 

agree that distances measured in one direction are positive and in the 
other direction are negative. Then, in a reference to a segment AB 
of the line, the order of the letters indicates the direction of the seg¬ 
ment, from A to B. The numerical value of AB is the number of units 
of length in t AB and its sign corresponds to its direction, + or —. To 
reverse the letters in a segment changes < —i—> _ i _ | _ | 

its sign. That is, a c b 

AB = - BA, or AB + BA = 0. Fig. 33 

Illustration 1. In Figure 33, the large arrow shows the positive direction. 
With the indicated unit, 

AB = 3; BA = - 3; BC = - 2; CA = - 1. 

* In this book, unless otherwise specified, the word number refers to a real number. 

56 




TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


57 


47. Rectangular coordinates. On each of the perpendicular axes 
OX and OF in Figure 34, we lay off a scale with the point 0, called 
the origin, as the zero on both scales. In the plane of OX and OF, 
we shall measure vertical distances in terms of the unit on OF and 
horizontal distances in terms of the unit on OX. Let P be any point 
in the plane. 

The horizontal coordinate, or the abscissa 

of P, is the perpendicular distance, x, from OF 
to P, where this distance is considered positive 
when P is to the right of OY and negative when 
P is to the left of OF. 

The vertical coordinate, or the ordinate of 

P, is the perpendicular distance, y, from OX to 
P, where this distance is considered positive 
when P is above OX and negative when P is 
below OX. 

The abscissa and ordinate of P, together, are 
called the rectangular coordinates of P. 

Illustration 1. In Figure 34, the coordinates of P are x = 4 and y = — 2. 
The coordinates of a point are usually written together within parentheses, with 
the abscissa first. Thus, we say that P is the point (4, — 2). In Figure 34, R 
is the point (— 3, 4). 

Note 1. Unless something is said to the contrary, the same unit of length 
will be used on the two axes in any coordinate system. 

The distance of a point P from the origin is called the radius vector 
of P. The radius vector is always considered positive when it is not 
zero. From triangle OPA in Figure 34, if r represents the radius 
vector, then 

r 2 - OA 2 + AP\ 

where OA and AP are the positive lengths of segments OA and AP. 
Regardless of the signs of x and y, OA 2 = x 2 and ^4P = y 2 . Hence, 
r 2 = x 2 + y 2 \ r - V* 2 + y\ (1) 

Illustration 2. In Figure 34, the radius vector of R is r = V9 + 16 = 5. 

The coordinate axes divide the plane into four parts called quad¬ 
rants which will always be numbered counter-clockwise, I, II, III, 
and IV, as in Figure 34. 

Illustration 3. A point R lies in quadrant II if the ordinate of R is positive and 
its abscissa is negative. 


Y 



p 


■ IV 

Fig. 34 










58 


TRIGONOMETRY 


EXERCISE 25 

Insert the proper sign, < or >, between the numbers: 

1. 2 and 5. 3. - 12 and 3. 5.-3 and 0. 7.-3 and - 5. 

2. 9 and 7. 4.-6 and 2. 6.-2 and 0. 8.-2 and - 7. 


Plot the following points on a coordinate system on cross-section paper: 

9. (3,4). 11. (6,1). 13. (3,0). 15. (1,-2). 17. (-3,5). 

10. (5,2). 12. (5,6). 14. (0,5). 16. (-3,2). 18. (4,-3). 

19. (-3,- 5). 21. (- 5,0). 23. (0,- 2). 25. (- 8,- 1). 

20. (- 2, - 1). 22. (0, - 3). 24. (- 3, 0). 26. (- 3, - 6). 


Plot each point and find the length of its radius vector: 

27. (3, 4). 30. (- 8, 15). 33. (7, - 24). 36. (- 6, - 8). 

28. (8, 15). 31. (- 4, 3). 34. (12, - 5). 37. (V3, - 1). 

29. (2, 3). 32. (- 3, 1). 35. (- 1, - 1). 38. (1, Vs). 

39. What is the abscissa of all points on the y- axis? What is the ordinate 
of all points on the z-axis? 

40. Describe the location of all points whose abscissa is 5; — 3; 0. 

41. Describe the location of all points whose ordinate is 7; — 2; 0. 

42. What is the abscissa of all points which lie on a line perpendicular to 
the z-axis at the point (3, 0)? 

43. What is the ordinate of all points which lie on a line perpendicular to 
the y- axis at the point (0, — 3)? 


48. The general angle. In elementary geometry and previously 
in this book, an angle was thought of merely as a ready-made figure 
consisting of two half-lines radiating from a common point; all angles 
were positive and between 0° and 360°. For many reasons, it proves 
convenient to generalize our previous notion of an angle as follows. 

Suppose that a half-line, radiating from a point O, rotates about O, 
either clockwise or counter-clockwise, from an initial position OA to a 
terminal position OB. Then, this rotation is said to generate an angle 
AOB, whose initial side is OA and terminal side is OB. The value of 
Z AOB is the amount of rotation used in generating the angle, where 
we agree to consider counter-clockwise rotation as positive and clock¬ 
wise rotation as negative. The description of an angle is incomplete 
until we are told the direction and amount of rotation * used in form¬ 
ing the angle. 

* As a measure of rotation, 1° is 1 /360th of a complete revolution. 


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


59 


Illustration 1. In Figure 35, the geometric figure AOB is the same in each 
diagram. In (1), A AOB — 45°. In ( 2 ), a complete revolution is indicated 
besides 45°: A AOB = 360° + 45° = 405°. In (3), the rotation is 45° less than 
360°, clockwise: A AOB = — 315°. In (4), A BO A = — 45°. In plane geometry 
we would have said that each angle in Figure 35 had the same value, 45°. 





Fig. 35 



Any number of complete revolutions, clockwise or counter-clock¬ 
wise, may be added to any of the rotations in Figure 35 without alter¬ 
ing the initial and terminal sides of the angles. Thus, we see that 
infinitely many 'positive and negative angles of unlimited numerical 
value correspond to any given pair of sides for the angles. 


Y 



49. Standard position of an angle. We 

shall say that an angle 6 is in its standard 
position on a coordinate system if the vertex 
of 6 is at the origin and the initial side of 6 
lies on the positive part of the horizontal axis. 

Illustration 1. To place 240° in standard position 
in Figure 36, imagine rotating OX about 0 from its 
initial position through 240° counter-clockwise to find 
the terminal side: 7 = 240°. Similarly we construct 
the other angles in standard position: 

a = - 60°; 6 = 30°. 


We shall say that an angle 6 is in a certain quadrant if the terminal 
side of 6 falls inside of that quadrant when d is in its standard position. 

Illustration 2. In Figure 36, 6 is in quadrant I 
and 7 is in quadrant III. Angles between 180° and 
270° are in quadrant III; angles between - 180° and 
- 270° are in quadrant II. An obtuse angle is one 
between 90° and 180°; hence, all obtuse angles lie in 
quadrant II. 

Two or more angles are said to be coter¬ 
minal if their terminal sides coincide after the 

angles are placed in their standard positions. 

« Fig. 37 

Illustration 3. In Figure 37, oc, p, and 7 are 
coterminal, a = 135°, = — 225°, and 7 = 360 + 135 — 495 . 











60 


TRIGONOMETRY 


EXERCISE 26 

(a) Sketch the angle in standard position and indicate the angle by a curved 
arrow. ( b ) Give the values of two coterminal angles, one positive and one 
negative, and indicate these angles in the figure by use of arrows. 


1. 

45°. 

5. - 60°. 

9. 180°. 

13. 495°. 

17. 

- 225°. 

2. 

90°. 

6. - 150°. 

10. 270°. 

14. - 390°. 

18. 

870°. 

3. 

120 

7. - 270°. 

11. - 90°. 

15. 0°. 

19. 

1050°. 

4. 

210 

8. - 315°. 

12. 420°. 

16. 135°. 

20. 

- 1200°. 

Draw a 

line from the origin 

i. in a coordinate system to the point 

P whose 


coordinates are given. Indicate by arrows, as in Figure 38, one positive and 



50. The trigonometric functions of any angle 0. Fig. 38 


Definition I. Place 6 in its standard position on a coordinate system. 
Choose any point P, not the origin, on the terminal side of 6; let (x, y) 
be the coordinates and let r be the radius vector of P. Then 





sin 0 
cos 0 
tan 0 
cot 0 
sec 0 
esc 0 


ordinate of P 

— - j or 

radius vector of P 

sin 0 

abscissa of P 
radius vector of P 

cos 0 

ordinate of P 
abscissa of P 

tan 0 

abscissa of P 
= > or 

ordinate of P 

cot 0 

radius vector of P 
abscissa of P 

sec 0 

radius vector of P 

s: - j or 

ordinate of P 

CSC 0 


y. 

r’ 

x 

r : 

y. 

x’ 

x 

y ] 

r 

x ; 

r 

V 


(1) 





















61 


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


Example 1. The terminal side of 0, in standard position, passes through 
(3, — 4). Find the functions of 0. 


Solution. 1. In Figure 40, 

r = Vx 2 + y 2 = V9 + 16 = 5. 

2 . Substitute (x = 3, y = - 4, r = 5 ) in ( 1 ): 


sin 6 = • 

-4 4 

5 5’ 

a 3 

COS 0 = x', 
o 

cot 0 = 

tan 0 — 

-4 4 

3 3 ; 

a 5 
sec 0 = 3 ; 

CSC 0 = 



The ratios in equations 1 are properly called Junctions of 6 because 
their values depend only on the value of 6 and not on the 'particular 
point P selected on the terminal side. For, suppose that we select any 
other point Pi with coordinates {x\, yf) and radius vector r%. Then, 
the new ratios formed with x h y h and ri will 
equal the corresponding ratios formed with 
x, y, and r, because (see Figure 41) triangles 
OAP and OA iP x are similar. 



Illustration 1 . In Figure 41, 


y± = y. 

n r’ 


= etc. 


Note 1 . In the left-most diagram in Figure 39, 0 is an acute angle. In the 
corresponding triangle OAP, y is the leg opposite to 0, x is the leg adjacent to 0, 
and r is the hypotenuse. Hence, on comparing pages 60 and 3, we see that 
Definition I gives the same values for the junctions of acute angles as the previous 
definition which held only for acute angles. This previous definition with its con¬ 
venient words adjacent side, opposite side, and hypotenuse must be absolutely 
discarded when 0 is not an acute angle. For, 0 can be thought of as an angle in 
a right triangle only when 0 is between 0° and 90°. 


51. Certain properties of the functions. We recall that the recip¬ 
rocal of a fraction a/b is 6/a. Hence, from Definition I we see that 


sin 0 = 


cos 0 = 


tan 0 = 


esc 0 


sec 0 


cot 0 

Illustration 1. 


CSC 0 = — 
sec 0 = 
cot 0 = 


sin 0’ 

1 

cos 0’ 

1 

tan 0 


1 1 r 1 , 

-—3 = -*=-, or ——3 = esc 0. 
sm 0 y y sin 0 


(1) 



















62 


TRIGONOMETRY 


If two angles are coterminal their trigonometric functions are 
identical, because Definition I involves only the terminal side of the 
angle. 

Illustration 2. Since 30° and 390° are 
coterminal, hence tan 30° = tan 390°. 

Note 1. The signs of the functions of 
angles in all quadrants are summarized in 
Figure 42. For any value of 0, sin 0 
and esc 0 have the same sign because 

sin 9 = — : • tan 0 and cot 0 have the 
esc 0’ 

same sign; cos 0 and sec 9 have the same 
sign. 

Example 1. Determine the signs of 
the functions in quadrant II. 

Solution. If 9 is in quadrant II, then, in Definition I, x < 0 and y > 0; r is 
always positive. Hence, regarding only signs, 

sin 0 = - = "T = +5 cos 0 = - = -^ = —; tan 0 = | = — = —. 

r + r + x — 

Example 2. Find the functions of 180°. 

Solution. 1. Draw 180° in standard position. The ordinate of any point on 
the terminal side is zero. 

2. Choose P as (— 3, 0). The length of OP is 3 units. Hence, r — 3. 

Y 3. Substitute r = 3, x = - 3, and y = 0 in the equa¬ 

tions of Definition I: 

tan 180° = = 0; sec 180° = ^g = — 1; 

sin 180° = jj = 0; cos 180° = ^ = - 1. 

Since y = 0, hence r/y and x/y are meaningless because division by zero has no 
meaning. Therefore, esc 180° and cot 180° are not defined; 180° has no cosecant 
and no cotangent. 

If 6 is any angle whose terminal side falls on a coordinate axis, then 
cot 6 and esc 6, or tan 0 and sec 6, are undefined for the reason met 
in discussing cot 180° and esc 180°. From Example 2 and problems 
in the next exercise, we find that a complete list of the undefined 
functions of angles from 0° to 360°, inclusive, is as follows: 

undefined j cot 0°; tan 90°; cot 180°; tan 270°; cot 360°; 1 

functions \ esc 0°; sec 90°; esc 180°; sec 270°; esc 360°. / 

For the present we are content merely to say that these functions are 
not defined, and we reserve this matter for later discussion. 



(-3.0) o 1 

Fig. 43 


sin 
C8C j 

| + 

all 

functions 


all 1 
others J 

1 - 

!+ 




O 


tan ) 
cot i 

+ 

C08 ) 

sec ) 

+ 

all ) 


all ) 


others 1 

1 — 

others 1 



Fig. 42 






TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


63 


EXERCISE 27 


Find the functions of 9 if the terminal side of 9, in standard position, passes 
through the given point: 


1. (4,3). 3. 

2. (5, 12). 4. 
11 . (- 6 , - 8 ). 

12. (1,5). 

13. (2,2). 


(7,24). 5. (12, 5). 7. (- 4, 3). 9. (7, - 24). 

(15, 8). 6. (3, 4). 8. (8, - 15). 10. (- 12, 5). 

14. (- 1 , - 1 ). 17. ( 1 , - Vs). 20 . ( 5 , 0 ). 

15. (2, - 5). 18. (- Vs, - 1). 21. (- 3, 0). 

16. (V3, 1). 19. (0, 3). 22. (0, - 4). 


Sketch the angle in standard position and find its trigonometric functions: 

23. 90°. 24. 270°. 25. 360°. 26. 0°. 27. - 90°. 28. - 180°. 


Determine the signs of the functions, to verify Figure 42. 

29. In quadrant I. 30. In quadrant III. 31. In quadrant IY. 

In what quadrants may 9 lie, under the given condition? 

32. sin 9 < 0. 34. tan 9 < 0. 36. sec 9 > 0. 38. sin 9 > 0. 

33. cos 9 < 0. 35. cot 9 < 0. 37. esc 9 < 0. 39. cos 9 > 0. 

In what quadrant must 9 lie, under the given condition? 

40. sin 9 < 0 and tan 9 > 0. 41. cos 9 < 0 and sin 9 > 0. 

Solution of Problem 40. 1. sin 9 < 0 in quadrants III and IV. 

2. tan 9 > 0 in quadrants I and III. 

3. Hence tan 9 > 0 and sin 9 < 0 only in quadrant III, which satisfies both 
Step 1 and Step 2. 

42. tan 9 < 0 and sin 9 < 0. 45. sec 9 < 0 and tan 9 > 0. 

43. cos 9 < 0 and cot 9 < 0. 46. esc 9 > 0 and cot 9 < 0. 

44. tan 9 > 0 and cos 9 < 0. 47. sec 9 < 0 and sin 9 > 0. 

48. Prove that, for all values of 9, sin 9 and cos 9 are numerically less 

than or at most equal to 1; sec 9 and esc 9 are numerically greater than or 
at least equal to 1. 

49. Fill in the remaining blanks in the following table: 


Angle 

sin 

cos 

tan 

cot 

sec 

CSC 

0° 







90° 

1 

0 

none 




180° 







270° 





















64 TRIGONOMETRY 

52. Functions of particular angles obtained geometrically. 

Example 1. Find the functions of 120° geometrically. 

Solution. 1. In Figure 44, right triangle AOP has 30° and 60° as its acute 

angles. Hence, from page 6 we recall that the hypotenuse, ^ 

OP, is twice as long as the shorter leg, OA. 

2. Select P so that OA = — 1 = x; then r = 2; 
y = V 3 . The student should complete the solution by sub¬ 
stituting in Definition I. 

Note 1. To obtain the functions of 225° we would use a 
point P on its terminal side with the coordinates (-1,-1). 

Example 2. Find all functions of 0 if sin 0 = f and 0 is in quadrant II. 

= 3 
r 

P in quadrant II for which y = 3 and r = 5. Sketch OP 
roughly to scale (Figure 45). 

2. From r 2 = x 2 + y 2 , x 2 = r 2 — y 2 . 
x 2 = 25 - 9 = 16. Hence x = — 4. 

3. Substitute in Definition I: cot 0 = — f; 

cos 0 = — sec 0 = — f; 

tan 0 = - f; esc 0 = f. 


p 1 



Vs 

\ 2 

-^120° 

60 v\ 


A 

L -1 

0 


Fig. 

44 


Solution. 


1 Since sin 0 = - = ^ > the terminal side of 0 goes through a point 
*• 5 



Fig. 45 


Comment. To construct OP accurately in Example 2, in Figure 46 draw a circle 
of radius 5 with center at 0. Draw a line parallel to OX and 3 units above it, 
intersecting the circle at P in quadrant II and at Q 
in quadrant I. .Then sin 0 = f if 0 is any angle 
whose terminal side is OP or OQ. Example 2 speci¬ 
fied that we should use OP. 


Except in special cases, if one function of 
an angle 6 is given we can find two different 
possible positions for the terminal side of 6, 
in standard position, and two corresponding 
values for 0 between 0° and 360°. Additional information is needed 
to determine which of the two values is desired in any problem. 



Illustration 1. Given that sin 0 = f, the two possible values for 0 are, ap¬ 
proximately, 0i = 40° and 0 = 140° in Figure 46. 

Note 2. In addition to the six trigonometric functions which we have defined, 
three other functions sometimes met are the versed sine, the coversed sine, and 
the haversine, which are defined as follows: 

vers 0 = 1 — cos 0; covers 0 = 1 — sin 0; havers 0 = 4(1 — cos 0). 


We shall not use these functions. 











TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


65 


EXERCISE 28 


Place the angle in standard position and find its trigonometric functions from 
the figure; rationalize denominators: 


1 . 

60°. 

5. 

135°. 

9. 

330°. 

13. 

- 150°. 

17. 495°. 

2. 

45°. 

6. 

225°. 

10. 

315°. 

14. 

1 

to 

H- 1 

O 

O 

18. 585°. 

3. 

210°. 

7. 

240°. 

11. 

- 135°. 

15. 

0 

cr 

00 

19. - 390° 

4. 

150°. 

8. 

0 

0 

0 

CO 

12. 

- 45°. 

16. 

- 330°. 

20. - 225°, 


The Roman numeral specifies the quadrant for 9. Construct 9 and find its 
other functions. 

21. tan 9 = A; 9 in (I). 27. 

22. cot 9 = f; 9 in (I). 28. 

23. cot 9 = - i; 9 in (II). 29. 

24. tan 9 = — 9 in (IV). 30. 

25. cos 9 = f; 9 in (IV). 31. 

26. esc 9 = f; 9 in (II). 32. 


sin 9 = — f; 9 in (III), 
cos 9 = — f; 9 in (II). 
sec 9 = f; tan 9 < 0. 
esc 9 = f; cos 9 < 0. 
tan 9 = f; sin 9 < 0. 
cot 9 = — cos 9 < 0. 


53. Graphical reduction to acute angles.* Trigonometric tables 
give the functions only of angles from 0° to 90°. Hence, if we are to 
use the tables for finding the functions of any other angle 6, in any 
quadrant, we must first express the functions of 6 in terms of the functions 
of some acute angle. 

Let 6 be an angle in standard position, in any quadrant. Let 

a be the acute angle between the terminal side of 6 and the x-axis. 
Then, we shall call a the reference angle for 6. 

Y Y Y 


0 = 140° 
a = 40° 




0 


0=-13O° 
a = 50° 


Theorem I. Any function of an angle 6, in any quadrant, is nu¬ 
merically equal to the same function of the reference angle for 6. That is, 
(any function of (?) = ± (same function of reference angle a), (1) 

where “ + ” or “ — ” is used according as the function of 6 is positive or 
negative. 

Illustration 1. If 9 = 240°, then a = 60°. From (1), 
sin 240° = — sin 60° = — 5 ^ 3 . 

* The instructor may omit this section if he prefers to treat this topic entirely 
by use of reduction formulas, as discussed later in Sections 54 to 59. 








TRIGONOMETRY 


66 


Proof of Theorem I. 1. Make the standard construction of Definition I 
for 0. Then, a is the acute angle AOP of right triangle AOP (see Figure 48). 


2. Let the positive numbers OA and AP be the 
lengths of OA and AP. Then x = ± OA and 
y = ±AP, where the signs “+” or depend 
on the quadrant in which P falls. 

3. By Chapter I, we may read the functions 
of a from triangle AOP ; the functions of 0 are 
found by Definition I: 


Y 



p 




e 

y 

>< 


A ^ 

o 


Fig. 48 


AP 

sin a = — J 


sin 0 



± sin a. 


tan a = 


AP. 

OA’ 


tan 0 = - 
x 


± AP 
± OA 



dz tan a. 


Similarly, we find that each function of 0 differs from the same function 
of a at most only in sign. 

Illustration 2. To find cos 118° 20', we notice from Figure 48 that the refer¬ 
ence angle is (180° - 118° 20') - 61° 40'. Since cos 118° 20' is negative, 

cos 118° 20' = - cos 61° 40' = - .4746. (Using Table VII) 


Example 1. Compute 378 tan 303° 50' by use of logarithms. 

Solution. 1. The reference angle is (360° — 303° 50') or 56° 10'. Also, we 

see that tan 303° 50' is negative. Hence, 378 tan 303° 50' = — 378 tan 56° 10'. 

2. First we compute the positive quantity 378 tan 56° 10': 

log 378 = 2.5775 (Table V) 

(+) log tan 56° 10' = 0.1737 (Table VI) 

log product = 2.7512. Hence, 378 tan 56° 10' = 563.9. 

3. Therefore, — 378 tan 56° 10' = — 563.9 = 378 tan 303° 50'. 

Note 1. The following rules are easily verified from figures: 

0 between 90 ° and 180 ° ; reference angle is a = 180 ° — 0 . ( 2 ) 

0 between 180 ° and 270 ° ; reference angle is a = 6 - 180 °. ( 3 ) 

6 between 270 ° and 360 ° ; reference angle is a = 360 ° — 0. ( 4 ) 


From (2), d = 180° - a; from (3), 0 = 180° + a; from (4), 
6 = 360° — a. Hence, from Theorem I we obtain the following 
equation, if a is acute 

[any function of (180° ± a), or of (360° - a)] = ± (same function of a). (5) 

In particular, since (180° - a) and a are supplementary angles, 
it follows from (5) that any function of an obtuse angle is numerically 
equal to the same function of the supplementary angle. 







TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


67 


If the numerical value of 6 is greater than 360°, before finding a 
function of 6 we may alter the angle by adding or subtracting an integral 
multiple of 360°. This alteration does not change the value of the 
function because the new angle obtained is coterminal with 6. 

Illustration 3. sin 398° = sin (398° — 360°) = sin 38°. 


EXERCISE 29 

Sketch the angle in standard position and find the reference angle: 

1. 139°. 4. 267°. 7. - 47°. 10. - 127°. 13. 535°. 16. - 278°. 

2. 171°. 5. 342°. 8. - 35°. 11. - 243°. 14. 680°. 17. - 390°. 

3. 232°. 6. 328°. 9. - 73°. 12. - 310°. 15. 460°. 18. - 530°. 


By use of a reference angle, find all functions of the given angle without 
using a table: 

19. 120°. 22. 240°. 25. 300°. 28. - 30°. 31. - 240°. 34. 570°. 

20. 135°. 23. 225°. 26. 315°. 29. - 45°. 32. - 210°. 35. 510°. 

21. 210°. 24. 330°. 27. 390°. 30. - 60°. 33. - 135°. 36. 420°. 


Find each function 

37. sin 118°. 

38. sin 125°. 

39. cos 127°. 

40. cos 171°. 

41. cot 227°. 

42. sec 211°. 

43. esc 315°. 

44. tan 260°. 


use of Table XI: 

45. cos 152° 17'. 

46. cos 172° 23'. 

47. tan 285° 12'. 

48. sin 324° 38'. 

49. cos 352° 3'. 

50. tan 211° 53'. 

51. sec 305° 19'. 

52. esc 143° 13'. 


53. sin (- 37° 15'). 

54. cos (-58° 38'). 

55. tan (- 137° 12'). 

56. cot (- 207° 42'). 

57. sin 439° 28'. 

58. tan 657° 49'. 

59. sin (— 727° 52'). 

60. cos (-416° 37'). 


or five-place logarithms: 


Compute by me of four-place 

61. 1.630 tan 112° 50'. 

62. 29.36 sin 147° 10'. 

63. .0937 cos 158° 10'. 

64. .1342 tan 118° 20'. 

65. 31.47 cos 313° 10'. 

66. 294.2 cot 298° 40'. 

Compute without using tables: 

73. 58 sin 270°. 76. 

74. 63 cos 180°. 

75. 27 sin 90°. 


67. .4314/tan 303° 36'. 

68. .003152/cos 243° 54', 

69. 7629/sin 321° 26'. 

70. 8625/cos 348° 16'. 

71. 152.3 cos 426° 15'. 

72. 613.5 tan 587° 41'. 


163 tan 360°. 
i7. .0613 sin 540°. 
78. .0139 cot 270°. 


79. .562 sin (- 180°). 

80. .127 cos (- 90°). 

81. .0013 esc (- 270°) 


68 


TRIGONOMETRY 


54. Reduction formulas. Certain relations exist between the func¬ 
tions of any two angles whose sum or whose difference is an integral 
multiple of 90°. We shall call these relations reduction formulas. 

Illustration 1. If 0 is any angle, then 0 and (360° + 0) are coterminal. Hence, 
we obtain the reduction formula 

[any function of (360° + 0)] = (same function of 0). (1) 


55. Functions of (— 0). If 6 is any angle, then * 

sin (— 9) = — sin 0; esc (—9) = — esc 9; 

cos (— 9) = cos 9; sec (— 9) = sec 0; (1) 

tan (— 0) = — tan 0; cot (— 0) = — cot 0. , 

Illustration 1. sin (— 27°) = — sin 27°; cos (— 27°) = cos 27°. 

Proof. 1. Let |8 = — 0. Make the standard construction of Defi¬ 
nition I for 0 and for /3, with P and Pi chosen on their terminal sides so 
that OP = OPi (see Figure 49). Angles XOP and XOP\ are numerically 
equal and r = n. Hence, PPi is perpendicular Y 

to OX, and is bisected by OX. Therefore, 
n = r; xi = x) yi = - y. 

2. From Definition I we obtain 


sin (— 0) = — = —- = — sin 0; 
v ' n r 

cos (— 0) = — = - = cos 0: etc. 
\ j Tl r ’ 



Note 1. The statements of the preceding proof hold if 0 is any angle in any 
quadrant, although Figure 49 pictures only a special case. In such a situation, 
the student should construct at least one other figure, for 0 in some other quadrant, 
and check through the whole proof for the new figure. 


EXERCISE 30 

Express the function in terms of a function of the negative of the angle: 

1. sin (-35°). 5. cos (- 128°). 9. esc (- 148°). 13. cos(-R). 

2. cos (- 75°). 6. cot (- 350°). 10. sec (- 250°). 14. sec (— H). 

3. tan (- 12°). 7. sec (- 35°). 11. tan (- 130°). 15. sin (- W). 

4. sin (-85°). 8. esc (- 245°). 12. cot (- 175°). 16. tan (— K). 

17. Repeat the proof of equations 1 by use of a figure like Figure 49 for 
the case where 0 is an acute angle. 

18. Repeat Problem 17, where 0 is an angle between 180° and 270°. 

* It is understood that certain formulas which we shall obtain and their proofs are 
ruled out automatically when the functions involved do not exist [recall (2), page 62.] 







TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 69 


66. Functions of (360° — 0).* The functions of (360° — 6 ) are 
the same as those of (- 6), because (- 6) and [360° + (- 0)] are 
coterminal. Hence, from Section 55, 

sin (360° — 6) = sin (— 6) = — sin Q. 

Similarly, we can prove the other formulas below. 

sin (360° — 0) — — sin 0; esc (360° — 0) = — csc0; j 

cos (360° — 0) = cos 0; sec (360° — 0) = sec 0; i (1) 

tan (360° - 0) = - tan 0; cot (360° - 0) = - cot 0. J 

Illustration 1. sin 342° = sin (360° — 18°) = — sin 18°. 


57. Functions of (90° + 0). If 6 is any angle, then 

sin (90° + 0) = cos 0; esc (90° + 0) = sec 0; j 

cos (90° + 0) = — sin 0; sec (90° + 0) = — esc 0; (1) 

tan (90° + 0) = — cot 0; cot (90° + 0) = — tan 0. J 

Illustration 1. Tan 115° = tan (90° + 25°) = — cot 25°. 


Proof. 1. Let /3 = 90° + 0. Make the standard construction for 0 
and for /3 with P and Pi chosen so that OP = OP i (see Figure 50). 

2. The acute angles AOP and A x PiO are equal. Hence, triangles AOP 
and A\PiO are congruent; the vertical side of either equals the horizontal 
side of the other. 

3. With attention to the signs of the coordi¬ 
nates, we obtain 

xi = — y; yi = x ; n = r. 

4. From Definition I, 

sin (90° + 9) = — = - — cos 9; 

' ' ' r \ r 

cot (90° + 9) = — = —^ = - tan 0; etc. 

2/i x 



58. Formulas for functions of (180° — 0), (0 ± 180°), (90° - 0), 

and (0 — 90°) can be proved by use of constructions like those in 
Figure 51. 





* In a minimum course, the instructor may desire to abbreviate the remainder 
of the chapter by merely calling attention to the formulas without consideration of 
proofs or exercises. 



















70 


TRIGONOMETRY 


sin (180° - 0) = sin 0; 
cos (180° - 0) = - cos 0; 
tan (180° - 0) = - tan 0; 

esc (180° - 0) = esc 0; 1 

sec (180° - 0) = - sec 0; 
cot (180° - 0) - - cot 0. J 

(1) 

[Any function of (0 ± 180°)] 

= ± (same function of 0). 

(2) 

[Any function of (90° — 0)] 

= (cofunction of 0). 

(3) 

[Any function of (0 — 90°)] 

= ± (cofunction of 0). 

(4) 


Note 1. On page 7, (3) was proved for any acute angle 0. We need to prove 
(2) only for (0 + 180°), or for (0 - 180°), because these angles are coterminal 
and hence have the same functions. For any function, in a particular case of 
(2) or (4), the ambiguous sign on the right can be determined by our knowledge 
of the signs of the functions in the various quadrants. 

★Note 2. After the reduction formulas for ( — 0) and (90° + 0) have been 
proved geometrically, all other reduction formulas can be obtained in sequence 
without geometrical reasoning. See Appendix, Note 3, for illustrations. 


59. S umm ary of reduction formulas. The following equations in¬ 
clude all possible reduction formulas as special cases. 


Any function of an 
[(even multiple of 90°) ± 0] 

Any function of an 
[(odd multiple of 90°) ± 0] 


=t (same function of 0). (1) 

± (cofunction of 0). (2) 


Illustration 1. 270° = 3 X 90°, an odd multiple of 90°. Hence, from (2), 


\_Any function of (270° dh 0)] = ± (cofunction of 0). (3) 


Reduction formulas may be used to accomplish the aims of Sec¬ 
tion 53, where reference angles were used. 

To express a function of an angle 0 in any quadrant in terms of a 
function of some acute angle, by use of reduction formulas, 

1. express as a sum or difference of the nearest multiple of 90° and an 
acute angle 0; 

2. use formula 1 or formula 2, with + or — on the right according as the 
desired function is positive or negative. 


Illustration 2. Sin 253° = sin (270° - 17°) = ± cos 17°, from (2). Since 
253° is in quadrant III where the sine is negative and since cos 17° is positive, 
we choose the minus sign: sin 253° = — cos 17°. 

Similarly, tan 253° = tan 17°. 


Illustration 3. Tan (- 165°) = tan (- 180° + 15°) = ± tan 15°, from (1). 
Since (— 165°) is in quadrant III where the tangent is positive, we choose the plus 
sign: tan (— 165°) = tan 15°. 

Similarly, sin (— 165°) = — sin 15°. 


TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 


71 


Note 1. To write a general formula which is a special case of (1) or (2) for 
any function, choose the proper sign in the right member by thinking of the special 
case where 9 is an acute angle. 

Example 1. Obtain a formula'for cot (0 + 270°). 

Solution. 1. By use of (2), cot (0 + 270°) = ± tan 9. 

2. If 9 is acute, (9 + 270°) is in quadrant IV and cot ( 9 + 270°) is negative, 
while tan 9 is positive. Hence, cot (9 + 270°) = — tan 9. 

Comment. The final formula in Step 2 is true for all values of 9. The special 
case, 9 acute, was used merely for convenience in determining the sign. 


EXERCISE 31 

Express all functions of each angle in terms of functions of an acute angle 
less than 45° by use of reduction formulas: 


1 . 

OO 

6. 

175°. 

11. 

324°. 

16. 

354°. 

21. 

- 136°. 

26. 

409°. 

2. 

55°. 

7. 

221°. 

12. 

352°. 

17. 

- 37°. 

22. 

- 157°. 

27. 

685°. 

3. 

127°. 

8. 

198°. 

13. 

195°. 

18. 

- 15°. 

23. 

- 247°. 

28. 

537°. 

4. 

132°. 

9. 

282°. 

14. 

237°. 

19. 

- 47°. 

24. 

- 265°. 

29. 

- 538° 

5. 

167°. 

10. 

267°. 

15. 

305°. 

20. 

- 58°. 

25. 

473°. 

30. 

- 467° 


Draw a figure for the case where 9 is an angle in quadrant I and prove 
the reduction formulas for the given angle by the method of Sections 55 and 57 : 

31. 180° - 9. 32. 180° + 9. 33. 9 - 180°. 34. 90° - 9. 35. 9 - 90°. 


36. 

tan (270° 

+ 0). 

37. 

cot (270° 

-0). 

38. 

sin (180° 

+ 0). 

39. 

cos (180° 

-0). 

40. 

tan (270° 

-0). 

41. 

sec (90° + 9). 

42. 

esc (90° + 0). 

43. 

tan (360° 

-0). 


tan (180° - 9). 
cot (180° + 9). 

46. sec (180° + 9). 

47. esc (360° - 9). 

48. sec (360° + 9). 

49. tan (540° + 9). 

50. sin (630° - 9). 

51. sin (9 - 360°). 


tan (450° — 9). 

53. tan (360° + 9). 

54. cos {9 - 360°). 

55. cos ( 9 — 450°). 

56. sec (- 270° - 9) 

57. esc (— 9 — 270°) 

58. sin (450° + 9). 

59. cos (540° — 9). 


Express in terms of a function of 9 by 

44. 

45. 


use of Section 59: 

52. 


SUPPLEMENTARY PROBLEMS 

By use of a figure, prove the reduction formulas for the given angle for 
the case where 9 is an angle in quadrant II. 

60. 90° + 9. 62. 270° - 9. 64. 180° + 9. 66. 9 - 90°. 

61. 360° - 9. 63. 270° + 9. 65. 180° - 9. 67. 90° - 9. 

68. Prove the reduction formulas for (270° + 9) and for (270° — 9) by 
the method discussed in the Appendix, Note 3. 


72 


TRIGONOMETRY 


REVIEW EXERCISE 32 

1. Tell the radius vector of each point: (6, 8); ( — 3, — 4). 

(a) Sketch the angle in standard position; ( b ) indicate by arrows, and tell 
the values of one positive and one negative angle coterminal with the given angle. 

2. 105°. 3. 240°. 4. 615°. 5. - 130°. 6. - 260°. 7. - 500°. 

Find the functions of 6 if the terminal side of 6, in standard position, passes 
through the given point: 

8. (- 3, - 4). 10. (- 7, 24). 12. (1, - V3). 14. (0, - 5). 

9. (5, - 12). 11. (5, - 5). 13. (3, 0). 15. (- V3, 1). 

In what quadrant does 9 lie under the given condition? 

16. sin 6 > 0; tan 9 < 0. 18. sin 9 < 0; sec 9 > 0. 

17. cos 9 < 0; tan 9 > 0. 19. esc 9 < 0; tan 9 > 0. 

The Roman numeral specifies the quadrant in which 9 lies. Find the other 
functions of 9 without using a trigonometric table. 

20. tan 9 = 9 in (III). 21. cot 9 = — 9 in (IV). 

Find the function without using a table: 

22. sin 315°. 25. tan 210°. 28. cos 135°. 31. cos (-30°). 

23. cos 225°. 26. cot 240°. 29. sin 330°. 32. sec (- 120°). 

24. sec 120°. 27. esc 300°. 30. tan (- 45°). 33. cot (- 60°). 

Construct each angle in standard position on a coordinate system and find 
its functions by direct use of Definition I, page 60 : 

34. 135°. 35. 210°. 36. 60°. 37. - 30°. 38. 90°. 39. - 180°. 

Find each function by use of Table XI: 

*40. sin 132°. 42. tan 217°. 44. tan 95° 17'. 46. sin (- 136°). 

41. cos 257°. 43. cot 319°. 45. sec 133° 24'. 47. cos (- 268°). 

Compute by use of four-place or five-place logarithms: 

48. 2.560 tan 157° 30'. 49. .9857 sin 312° 27'. 

Express * in terms of a function of 9: 

50. sec (180° + 9). 52. cos (9 - 90°). 54. cot (360° + 9). 

51. sin (270° - 9). 53. tan (9 + 540°). 55. tan (720° - 9). 

56. Construct 9 and (90° + 9) in standard position on a coordinate 
system, with 9 as an angle between 180° and 270°. From the figure, prove 
the reduction formulas for functions of (90° + 9). 

* The remaining problems should be omitted by a class which has not considered 
Sections 56 to 59. 


CHAPTER V 
RADIAN MEASURE 

60. Radian measure. Previously in this book, angles have been 
measured in degrees and their fractional parts. Hereafter, we shall 
frequently use an angular unit called a radian which has important 
applications, particularly in more advanced mathematics. 


Definition I. One radian is the measure of an angle which, if its 
vertex is 'placed at the center of a circle, subtends on the circumference an 
arc equal to the radius of the circle. 

Illustration 1. In Figure 52, Z BOC is 1 radian. 

Hence, the length of the subtended arc BC is r. 

Since an arc of length r subtends an angle 
of one radian at the center, hence * the whole 
circumference, whose length is 27r X r, sub¬ 
tends 2t X (one radian) or 2ir radians. 

Since the whole circumference subtends 
360°, hence 360° = 2t radians, or 

180° = 7 r radians; 



1° = ^ radians = .0174533 radians; 
180 


( 2 ) 


1 radian =- = 57.2958°, approximately. (3) 

7r 

Thus, to change from degrees to radians, multiply the number of 


degrees by ^ • To change from radians to degrees, multiply the 
180 

number of radians by —— • 

7T 

5t r ,. 5t r 180° _ ., no 

Illustration 2. -g- radians — -g- • — 15U . 

f 7 r radians = f(180°) = 270°, because r radians equal 180°. 

90 ° = 90 • — radians = %. radians. When convenient, we shall leave a value 
180 2 

in radians expressed in this way as a multiple of tt. In decimal form, 

90° = \tt = R3.1416) radians = 1.5708 radians, (t = 3.14159 • • •) 

* From geometry we know that angles at the center of a circle are proportional 
to their subtended arcs. 


73 



74 


TRIGONOMETRY 


EXERCISE 33 

Express each angle in radian measure: 

1. 60°. 4. 135°. 7. 210°. 10. 270°. 13. 36°. 16. - 180°. 

2. 30°. 6. 150°. 8. 225°. 11. 300°. 14. 126°. 17. - 360°. 

3. 45°. 6. 120°. 9. 240°. 12. 330°. 15. - 90°. 18. - 540°. 

Given the radian measure of an angle; change to degrees: 


19. 

7T 

22. 

X 

26. %-■ 

28. 

7 x 

31. 

— X. 

37. 

7 x 

6' 

9 ’ 

4 


15 

32. 

3x. 


~ ~9~ 

20. 

7T 

23. 

7T 

26. ¥-■ 

29. 

llx 

33. 

2. 

38. 

13x 

3’ 

2* 

6 


15 

34. 

4. 


9 


7r 

24. 

3x 

2 x 

30. 

5x 

35. 

2.5. 

39. 

~T2 x. 

21. 


27. — 


36. 



4’ 

T 

3 

12* 

3.6. 

40. 

T7T. 

Construct each angle approximately to scale: 





41. 

3 radians. 


42. f radians. 

43. 4 radians. 

44. 

2 radians. 


45. In a triangle, one angle is 36° and another is r radians. Find the 
third angle in radians. 

46. Through how many radians does the hour hand of a clock revolve 
in 40 minutes? 

47. Through how many radians does the minute hand of a clock revolve 
in 25 minutes? 

Note. Hereafter, when no unit is indicated in giving the value of an angle, 

7T \/2 

the unit is understood to be a radian. Thus, sin -g = sin 45° = • 

Give the value of each function, without using a table: 


48. 

. X 

Sin 3' 

52. cot^ - 

56. 

sin x. 

. llx 

60. sin — ^— 

6 

49. 

X 

cos 6 

53. sin 

57. 

cos 2x. 

61. tan 

b 

50. 

X 

tan 4 * 

54. cosh¬ 

58. 

5x 

tan T' 

62. cos(-f)- 

51. 

2x 

sec- 3 - 

er 3x 

55. cos -y- 

59. 

cot 

4 

63. rin(-f)- 

Express in radians, computing decimal values by 

use of Table Y: 

64. 

38° 21'. 

65. 123° 50'. 

66. 

273° 45'. 

67. 183° 18'. 

Hint. 21' is 21/60 of 1° or .35°. 

Hence, 

38° 21' = 

38.35°. 

68. 

347° 39'. 

69. 284° 53'. 

70. 

141° 49.6' 

'. 71. 315° 17.4'. 


RADIAN MEASURE 


75 


Express * each function in terms of a function of 9: 


72. 

sin (t — 6). 

74. 

tan (27t — 9). 

76. 

sin (27r + 9). 

73. 

COS (-7T 6). 

75. 

cos (2x — 9). 

77. 

cot (2ir + 9). 

78. 

ten (| + e) : 

80. 

»(£ + •)■ 

82. 

csc (3E - «) 

79. 


81. 

“(I-*)' 

83. 

sec (t + 6 ) 


61. Relation between arc, angle, and radius. In a circle of radius r, 
in Figure 53, let s represent the length of the arc subtended on the 
circumference by a central angle of 6 radians. Since 1 radian at the 
center subtends an arc whose length is r, hence 6 radians subtend an 
arc whose length is d • r. That is, 

s = rO ; (1) 

or, arc = (radius) X (angle, expressed in radians). (2) 


Illustration 1. If r = 25 feet and s = 75 feet, 
s 75 

then, from (1), 0 = - = — = 3 radians. 

r Zo 

Example 1. Find the length of the arc sub¬ 
tended by a central angle of 35° in a circle whose 
radius is 20 feet. 

Solution. 1. Find the radian measure of 35°: 

e = 35 ' llo = S radians - 

2. Since r = 20, hence 

S = r0 = 2og = ^, or s = 12.2 ft. 



★62. Linear and angular velocity. Let v represent the linear 
velocity, or rate of motion, of an object P which is moving with 
uniform velocity on the circumference of a circle whose center is 0 
and radius is r. The velocity v is the length of arc passed over hy P in 
one unit of time. Let co (called omega ) represent the angle through 
which OP turns about 0 in one unit of time; we 
call co the angular velocity of P. If co is measured 
in radians, then, from (1) of Section 61, 

v = rw. (1) 

If, in t units of time, P moves over an arc of 
p length s and OP revolves through angle 6, then 
s = vt; 0 = cot. (2) 

* These problems should be omitted if Sections 56 to 59 were omitted. 






76 


TRIGONOMETRY 


Note 1. All motions considered will be at uniform speed. 

Example 1. If P moves 38 feet in 4 seconds on the circumference of a 
circle whose radius is 6 feet, find the angular velocity of P. 

19 

Solution. 1. From s = vt, 38 = 4i>; v = ft. per sec. 

2. From (1), ^ = 6o>; co = ^ radians per sec. 

Z 1Z 

Example 2. A belt passes over the rim of a flywheel, 30 inches in di¬ 
ameter. Find the speed of the belt if it drives the wheel at the rate of 
5 revolutions per second. 

Solution. 1. A point on the belt moves with the same linear velocity as a 
point on the rim of the wheel. 

2. One revolution equals 2 ir radians; hence, co = 5 X 27r = 107T radians. 

3. From v — rco, v = 15(10tt) = 1507T = 471.2 ft. per sec. 

EXERCISE 34 

Obtain each result in this exercise to three significant figures. 

By use of s = rd, find whichever of (r, s, 9) is not given; a represents the 
degree measure corresponding to 6 radians: 


1. 

r 

= 

10 ft.; 9 

= 2.3. 

10. 

r 

= 

450 ft.; 

9 

= 5.7. 

2. 

r 

= 

25 ft.; 9 

= 1.6. 

11. 

s 

= 

175 in.; 

r 

= 4 ft. 

3. 

9 

= 

3.5; s = 

20 ft. 

12. 

s 

= 

2500 ft.; 

r 

= .75 mi. 

4. 

9 

= 

fx; s — 

150 ft. 

13. 

a 

= 

120°; s 

= 

375 in. 

5. 

9 

= 

lx; s = 

125 in. 

14. 

a 

= 

240°; s 

= 

30.5 in. 

6. 

9 

= 

6.8; s = 

50.6 in. 

15. 

a 

= 

135°; r 

= 

16 in. 

7. 

s 

= 

38.5 ft.; 

r = 16 ft. 

16. 

a 

= 

110°; r 

= 

25 in. 

8. 

s 

= 

19.2 ft.; 

r = 12.6 ft. 

17. 

a 

- 

215°; r 

= 

30 ft. 

9. 

r 

= 

200 ft.; 

9 = 3.2. 

18. 

a 

= 

340°; r 

= 

50 ft. 


19. If an arc, 30 feet long, subtends an angle of 2 radians at the center 
of a circle, find its radius. 

20. If an arc, 120 feet long, subtends an angle of 3.5 radians at the center 
of a circle, find its diameter. 

21. In a circle, 16 inches in diameter, how long an arc is subtended by an 
angle of 2.4 radians at the center? 

22. Through what angle in radians would a runner turn in going 
100 yards on a circular track 140 yards in diameter? 

23. The angle between two tangents to a circle from a certain point is 
1.3 radians. Find the shortest distance along the circumference between 
the points of tangency, if the radius of the circle is 20 inches. 


RADIAN MEASURE 


77 


24. A railroad curve, in the form of an arc of a circle, is 850 yards long. 
If the radius of the circle is 950 yards, find the angle in degrees through 
which a train turns in going around the curve. 

25. In going around a circular curve which is 645 yards long, a railroad 
train turns through an angle of 57° 35'. Find the radius of the curve. 

26. At the intersection of two streets, a street-car track accomplishes a 
change of direction of 36° 25' by following the arc of a circle for 185 feet. 
Find the radius of this arc. 

SUPPLEMENTARY PROBLEMS 

27. A right circular cone is formed by cutting a sector of 130° out of a 
circle whose radius is 40 inches and by then bringing the edges of the sector 
together. Find the radius of the base of the cone. 

28. The base of a right circular cone has a diameter of 25 feet and its 
slant height is 40 feet. The surface of the cone is cut along a straight line 
from its vertex to a point on the base, and the surface is then spread out 
flat to form a sector of a circle. Find the angle of this sector in degrees. 

29. Find the linear and angular velocities of a man who runs 1.15 times 
around the circumference of a circle, 68 feet in diameter, in 9 seconds. 

30. A wheel, 3 feet in diameter, makes 420 revolutions per minute. 
Find the linear velocity of a point on the rim in feet per second. 

31. Find the angular velocity in radians per minute of the end of 
(a) the minute hand of a watch; (6) the hour hand. 

32. The 400-meter race in the 1932 Olympic Games was won by William 
Carr of the United States in the new world’s record time of 46.2 seconds. 
Assuming that he ran with uniform speed on a circular track 130 meters in 
diameter, find his linear velocity in miles per hour and his angular velocity 
in radians per second. Given that one mile equals 1609 meters. 

33. Assume that the earth is a sphere with a radius of 4000 miles which 
revolves on its axis once in 24 hours. Disregarding the other motions of 
the earth, find the linear velocity, per second, of a point on the equator. 

34. Find the radius of a flywheel if it is turned at 1250 revolutions 
per minute by a belt moving over the rim of the wheel with a velocity of 
60 feet per second. 

35. A flywheel, 48 inches in diameter, is driven by a belt moving over 
the rim with a velocity of 3600 feet per minute. Find the angular velocity 
of the wheel (a) in radians per minute; (6) in revolutions per minute. 

36. A moving belt drives two flywheels whose diameters are 30 inches 
and 50 inches. If the smaller wheel turns through 280 revolutions per 
minute, find the velocity of the belt in feet per second and the angular 
velocity of the larger wheel in revolutions per minute. 


CHAPTER VI 

VARIATION AND GRAPHS OF THE FUNCTIONS 


63. Periodicity of the functions. Let 9 be a variable angle, free 
to assume any value. If 9 varies from one value to another, for 
instance from 0° to 1080°, then the terminal side of 9, in standard 
position, revolves about the origin of the coordinate system. The 
positions of the terminal side of 9, and hence the values of its trigo¬ 
nometric functions, repeat themselves periodically at intervals of 360° 
in the values of 6. For every value of 6, the angles 6 and (360° + 6) 
are coterminal and hence their functions are identical. For this reason 
we say that any trigonometric function of 6 is a periodic function of 6 
with 360°, or 2i r radians, as a period. On account of this periodicity, 
all information about the values of the trigonometric functions can 
be obtained by considering only angles from 0° to 360°. 

Illustration 1. For all values of 9, sin ( 9 360°) = sin 9. For instance, 

§ = sin 30° = sin (30° + 360°) = sin (30° + 720°) = • • • ; the value § recurs 
'periodically as a value of sin 9 at intervals of 360° in the values of 9. 

From Chapter IV, we recall that tan (6 + 180°) = tan 6 and 
cot (6 + 180°) = cot 6. Hence, tan 6 and cot 6 have 180° as a period, 
as well as 360°. If d varies, the values of tan 6 and cot 0 repeat peri¬ 
odically at intervals of 180° in the values of 6. Therefore, we can 
obtain all information about tan 6 and cot 9 by restricting ourselves 
to angles from 0° to 180°. 

★Note 1. By a more advanced discussion it could be proved that the smallest 
period of tan 9 and cot 9 is 180°, and of all other functions is 360°. 

★Note 2. Suppose that a variable w is a function of a variable z, where the 
word function does not necessarily mean trigonometric function. Then, w is said 
to be a periodic function of z with p as a period in case identical values of w cor¬ 
respond to any two values of z which differ by p. 

64. Variation of sin 6 and cos 6. In Figure 55, 9 is in standard 

position and OP = 1 — r. Hence, sin 9 = j = y and cos 9 = x. 

That is, the abscissa and ordinate of P equal cos 9 and sin 9, respec¬ 
tively. If 9 increases from 0° to 360°, P moves just once around the 

78 


VARIATION AND GRAPHS OF THE FUNCTIONS 


79 


Y 



circumference of the circle whose 
radius is 1 and center is 0. The cor¬ 
responding changes in cos 9 and sin 9 
can be observed by noticing the varia¬ 
tions in the coordinates of P. 

Illustration 1. If 9 increases from 0° to 
90°, then P moves from Pi to Pi and the 
abscissa of P, or cos 9, decreases from 1 to 0. 
If 9 increases from 90° to 180°, cos 9 de¬ 
creases from 0 to — 1. Similarly, the student 
should describe all facts about sin 9 and 
cos 9 which are summarized in the table on 
page 81. 


65. Variation of tan 6 . In Figure 56, for 9 < 90°, P is chosen so 


that x = 1. Then tan 9 = - = % = y. If 9 ap- 

x 1 

proaches 90°, P moves up beyond all limits on the 
line perpendicular to OX at A, and hence the 
ordinate of P, or tan 9, increases without bound. 
Therefore, tan 9 is greater than any specified num¬ 
ber, however large, if 9 is sufficiently near to 90°. 
For brevity, we say that tan 6 becomes posi¬ 
tively infinite as 6 approaches 90°, if 6 < 90°. 



Illustration 1. Tan 89° = 57; tan 89° 59' = 3438; tan 89° 59' 59" = 206,265. 


Figure 56 does not apply if 9 = 90°. Then, in Definition I, 
page 60, P is on the y- axis and x = 0. Hence, tan 90° is not defined 
because y/x is meaningless if x = 0. 

In Figure 57, for Q > 90°, x = - 1: 

tan 9 = = - y; or, y = - tan 9. 

By a discussion like that for 9 < 90°, we find 
that tan 6 becomes negatively infinite if 9 ap- 
■x proaches 90°, if 9 > 90°. To abbreviate all the 
facts about tan 9 for 9 near 90°, we write 
Fig. 57 tan 90° = oo, (1) 

which is read tan 90° is infinite. The symbol “ °o,” abbreviating in¬ 
finite or infinity, is not a number and hence (1) is not an equation in 
the ordinary sense of the words. We use (1) as a symbolic abbrevia¬ 
tion for the following two statements: 















80 


TRIGONOMETRY 


A. There is no tangent for 90°. 

B. If 0 < 90°, tan 0 is greater than any specified number, however 
large, if 0 is sufficiently near to 90°. If 0 > 90°, tan 0 is less than any 
negative number, however large its numerical value, if 0 is sufficiently 
near to 90°. 

From Figure 56, if 9 increases from 0° to 90°, then tan 9 starts 
with the value tan 0° = 0 and increases through all positive values, 
without bound; for brevity, we say that tan 9 increases from 0 to 
+ oo (read plus infinity). In Figure 57, if 9 decreases from 180° to 90°, 
then tan 9 starts with the value tan 180° = 0 and decreases without 
bound through all negative values. Hence, if 9 increases from 90° to 
180°, tan 9 increases from — oo to 0. 

Note 1. The variation of tan 6 from 6 = 180° to 0 = 360° is the same as from 
0 = 0° to 6 = 180°, because tan 6 repeats its values periodically at intervals of 
180° in the values of 6. In particular, we have tan 270° = oo. 

66. Variation of cot 9 . By means * of Figure 58, with reasoning 
like that for tan 9 , we could show that 
cot 0° = oo . Then, since the values of 
cot 9 repeat periodically at intervals of 
180° in the values of the angle 9 , it follows 
that cot 180° = cot 360° = oo . The stu¬ 
dent should verify the statements about 
cot 9 in the table on page 81. 

Note 1. The variation of cot 0 can also be 

learned by recalling that cot 0 = • Hence, 

if 0 varies so that tan 0 increases in numerical value, then the numerical value of 
cot 0 decreases, and vice versa. When tan 0 approaches zero, cot 0 becomes infinite, 
and when tan 0 becomes infinite, cot 0 approaches zero. 

67. Variation of sec 0 and esc 0. We know that sec 9 = ^ 

cos 9 

If 9 increases from 0° to 90°, then cos 9 decreases from 1 to 0; hence 
sec 9 starts at the value sec 0° = 1 and increases without bound. 
We recall that 90° itself has no secant: 1/cos 90°, or 1/0, has no mean¬ 
ing. If 9 increases from 90° to 180°, cos 9 decreases from 0 to — 1; 
hence sec 9 is negative and decreases from — » to — 1. 

* See Appendix, Note 4, for a composite diagram exhibiting line representations 
for all functions of an angle in any quadrant. 








VARIATION AND GRAPHS OF THE FUNCTIONS 


81 


Note 1. The student should complete the verification of the items for sec 9 
and esc 9 in the following table. We notice that the symbolic values °o for 
sec 9 and esc 9 correspond to the zero values of cos 6 and sin 9, respectively, 

because sec 9 = ——- and esc 9 = — 

cos 9 sin 9 

sec 90° = oo ; sec 270° = oo ; esc 0° = oo ; esc 180° = oo. 

Note 2. In the following summary, D and I abbreviate decreases from and 
increases from, respectively. For instance, we read that if 9 increases from 180° 
to 270°, then cos 9 increases from — 1 to 0. 


9 Incr. 
^''^Fbom 
Then 

0° to 90° 

90° to 180° 

180° to 270° 

270° to 360° 

sin 6 

I. 

0 to 1 

D. 

1 to 0 

D. 

0 to — 1 

I. 

- 1 to 0 

cos 6 

D. 

1 to 0 

D. 

0 to — 1 

I. 

- 1 to 0 

I. 

0 to 1 

tan 0 

I. 

0 to + oo 

I. 

1 

8 

o’ 

o 

I. 

0 to + oo 

I. 

— oo to 0 

cot 9 

D. 

+ oo to 0 

D. 

0 to — oo 

D. 

+ oo to 0 

D. 

0 to — oo 

sec 9 

I. 

1 to + OO 

I. 

— oo to — 1 

D. 

— 1 to — oo 

D. 

4- oo to 1 

esc 9 

D. 

+ OO to 1 

I. 

1 to + 00 

I. 

— oo to — 1 

D. 

— 1 to — oo 


EXERCISE 35 

Describe the variation of each function of 9 if 9 increases from the first 
angle to the second: 

1. 0° to 360°. 3. - 90° to 270°. 5. - 90° to 90°. 7. tt to 3 tt. 

2. 360° to 720°. 4. - 180° to 0°. 6. - 2tt to 0. 8. - tt to tt. 

Write the two statements like A and B, page 80, which are abbreviated by 
the given symbolic equation: 

9. tan 270° = 11. esc 180° = oo. 13. cot 360° = «. 

10. sec 270° = =o. 12. cot 180° = oo. 14. esc 0° = oo. 


SUPPLEMENTARY PROBLEMS 


15. In Figure 56, page 79, prove that OP = sec 9; in Figure 57 prove 
that OP = — sec 9. Then write a complete discussion like that of Sec¬ 
tion 65 to show that sec 90° = <*>. 

16. By use of Figure 58, write a complete discussion to show that 
cot 0° = oo and esc 0° = °° . 

17. Suppose that we know only the variation of tan 9. By use of 

the fact that cot 9 = —> find how cot 9 varies as 9 increases from 0° 
tan 9 


to 180°. 

18. Prove the facts stated in the Appendix, Note 4, equations 1. 






















82 


TRIGONOMETRY 


68. A change of notation. Hereafter, only rarely will it be nec¬ 
essary to think of angles placed in standard position and, unless 
otherwise specified, the letters x and y will no longer represent the 
coordinates of a point on the terminal side of an angle. 

69. Graph of sin x and cos x. Let x represent a variable angle, 
positive, or negative, or zero. Let y = sin x; then, for every value 
of x there exists a corresponding value of the related variable y. 
Each pair of corresponding values of x and y can be considered as the 
coordinates of a point in a system of rectangular coordinates. Then, 
the locus of all such points is called the graph of the function sin x } 
or the graph of the equation y = sin x. Similarly, we can consider 
obtaining the graph of cos x, or of any other function of x. 

Y 



Illustration 1. In the following table, the entries for angles from 0° to 90° were 
obtained from Table IY or from memory. To obtain the entries from 90° to 
180°, we recalled that sin 105° = sin 75°, sin 120° = sin 60°, etc. The graph of 
sin x in Figure 59 was obtained by extending the table from — 360° to 360°, 
and by plotting the corresponding points (ay y,), where x represents the radian 
measure of the angle. The graph of cos x in Figure 60 was obtained similarly. 


x {degrees) 

0° 

30° 

45° 

60° 

75° 

90° 

105° 

120° 

135° 

150° 

180° 

x ( radians ) 

0 

.52 

.79 

1.05 

1.31 

1.57 

1.83 

2.09 

2.36 

2.62 

3.14 

y = sin x 

0 

.50 

.71 

.87 

.97 

1.00 

.97 

.87 

.71 

.50 

0 























VARIATION AND GRAPHS OF THE FUNCTIONS 


83 


The graph of any trigonometric function of x, in particular the 
graph of sin x or of cos x, consists of infinitely many repetitions, to 
the right and to the left of the origin, of that part of the graph obtained 
as x varies from 0° to 360°. This is true because each of the functions 
has 360° or 2 tv radians as a period. 

Note 1. It is seen that the graphs of sin x and cos x have the same shape but 
differ in their locations with respect to the axes. This can be proved by rec alling 
that sin ( x + 90°) = cos x; that is, the cosine of any angle x equals the sine of 
an angle which is 90° greater. Hence, if we shift the graph of sin x to the left, 
parallel to the x-axis, through a distance equal to 7t/2 radians, or 90°, of the 
x-scale, we obtain the graph of cos x. 

Unless otherwise directed by the instructor, in graphing a trigo¬ 
nometric function of x, use the same unit of length on the scales on 
the two coordinate axes, and suppose that x is measured in radians.* 

Illustration 2. This agreement was followed in Figures 59 and 60, and in the 
large-scale graph of y = sin x from 0 to 7T in Figure 61; the fundamental values 
on the x-axis, 7t/2, tv, 3it/2, etc., do not fall on main division marks on the x-scale. 
In Figure 61, we plot (x = 0, y = 0), (x = .52, y = .50), etc., which are obtained 
from the preceding table of values for sin x. The following decimals are useful 
in labeling the x-scale. 


Fraction 

iTT 

iTT 

£tt 

Att 

iTT 

TT7T 

|7T 

iT 

|7T 

7T 

Decimal Value 

.52 

.79 

1.05 

1.31 

1.57 

1.83 

2.09 

2.36 

2.62 

3.14 


Y 



Fig. 61. y — sin x 

70. Graphs of tan x, cot x, sec x, and esc x. We recall that there 
is no tangent for 90° or for any angle differing from 90° by an integral 
multiple of 180°. This causes the breaks in the graph in Figure 62, on 
page 85, at x = and x = f 7r. Since tan x is a periodic function 
of x with a period of 180°, or 7r radians, the graph of tan x consists of 
infinitely many separated branches like the one from x = to x = f 7r. 

* In a course not leading to calculus, the instructor may desire to permit the use 
of degree measure for x and the arbitrary selection of the scale on the x-axis. 



























84 


TRIGONOMETRY 


Illustration 1. Later, the student will verify the graphs on page 85. In Fig¬ 
ure 62, the graph of y = tan x from x = 0 to x = |x was constructed by use of 
the following table which was made up from Table IV. In forming a similar 
table for values of x from 180° to 360°, we would make use of the fact that 
tan (9 + 180°) = tan 9, 

for all values of 9. In Figure 62, the broken lines, called asymptotes, do not touch 


x ( degrees) 

0° 

15° 

30° 

45° 

60° 

75° 

78° 

CO 

o 

o 

x ( radians ) 

0 

T2 7T 

in 

ix 

hr 

JjTT 

1.36 


y = tan x 

0 

.3 

.6 

1.0 

1.7 

3.7 

4.7 

-f 00 


the graph because there is no tangent, and hence no point on the graph, corre¬ 
sponding to x = £x or x = fx. As x approaches x/2 from the left, the point on 
the graph whose abscissa is x approaches the corresponding broken line and the 
ordinate of the point increases without hound. We can approach as closely as we 
desire to either broken line, but never reach the line, by receding sufficiently far 
from the x-axis on the graph. In Figure 62, the asymptotes were drawn first to 
aid in the construction of the graph. 

Note 1. The graph of a function of x (not necessarily a trigonometric function) 
is said to be discontinuous where there is a break in the graph. Thus, tan x and 
sec x are discontinuous at x = \rr, x = |7r, etc., while cot x and esc x are discon¬ 
tinuous at x = 0, x = 7r, etc. 

The student should practice quickly reproducing the graphs of 
the trigonometric functions roughly to scale by use of only a few 
plotted points. From the graph of a function we can easily recall 
important facts about its values. 

Illustration 2. From the appropriate graphs, we read the following facts. 
The values of sin x and of cos x all fall between — 1 and + 1, inclusive, because 
no ordinate on the graphs of the functions exceeds 1 in numerical value. The 
numerical value of sec x or of esc x never is smaller than 1. As x decreases from 
7T to 7t/2, tan x decreases from 0 to — <». 


EXERCISE 36 


Make a table of values of the function for the indicated range of values of x, 
using at least five values of x in each interval of 90°. Then graph the function 
on cross-section paper. Draw any asymptotes which the graph possesses. 


1 . 

2 . 


3. 

4. 


5. 


6 . 


y = cos x; 
y = sin x; 
y = cot x; 
y = tan x; 
y — sec x; 
y = esc x; 


from x = 0 to x = §7r. 
from x = — x to x = x. 
from x = 0 to x = 3x. 
from x = — fx to x = fx. 
from x = — f x to x = fx. 
from x = 0 to x = 2x. 
















VARIATION AND GRAPHS OF THE FUNCTIONS 


85 




Fig. 62. y = tan x Fig. 63. y = cot x 




Fig. 64. y — sec x 


Fig. 65. y = esc x 
































86 


TRIGONOMETRY 


By inspection of the proper graph, tell all values of x between — 360° and 


360 c 

inclusive ,, 

for which 

the equation is 

true: 








7. 

sin x 

= 0 . 

11. 

tan 

X 

= 0 . 


15. 

sin 

x = 

- 

1 . 

19. 

tan 

x = 

OO 

8. 

sin x 

= 1 . 

12. 

CSC 

X 

= - 

1 . 

16. 

cos 

x = 

- 

1 . 

20. 

cot 

x - 

00 

9. 

cos X 

= 1 . 

13. 

sec 

X 

= - 

1 . 

17. 

sec 

x = 

1 . 


21. 

CSC 

x = 

OO 

10. 

cos X 

= 0 . 

14. 

cot 

X 

= 0 . 


18. 

CSC 

x = 

1 . 


22. 

sec 

x = 

OO 


Graph the two functions from x = — 2ir to x = 2-ir on the same coordinate 
system, by use of the values of the functions only for 0°, 45°, and other angles 
differing from these by multiples of 90°: 

23. sin x and esc x. 24. cos x and sec x. 25. tan x and cot x. 

Comment. Any ordinate of the graph of esc x is the reciprocal of the corre¬ 
sponding ordinate of sin x because esc x = • This fact could be used to con¬ 

struct the graph of esc x geometrically by use of only the graph of sin x. Similar 
remarks apply to tan x and cot x, and to sec x and cos x. 

26. By inspecting Figures 59, 60, 62, 63, 64, and 65, describe the variation 
of each of the trigonometric functions of x, if x decreases from 2r to 0 . 

★71. Graphs of other functions. 

Example 1. Graph the function y = sin 3x from x — 0 to x = 2ir. 

Solution. 1. If x increases from 0 to 27T, then 3x increases from 0 to 67 t, and 
hence sin 3 x passes through all values which would be obtained for sin x if x 
varied from 0 to 67 r. Thus, the graph of sin 3x will exhibit three waves whereas 
the graph of sin x shows only one wave in the interval from x = 0 to x = 2tt- 

2. The zero values of sin x occur at intervals of 7 T radians in the values of x. 
Hence, the zero values of sin 3x occur at intervals of 7 r radians in the values 
of 3x, or at intervals of 7 t/3 radians in the values of x. This suggests the values 
of x used in the table on page 87, from which the principal points in the graph in 
Figure 66 were determined. 

Y 



Fig. 66 . y = sin 3x 







VARIATION AND GRAPHS OF THE FUNCTIONS 


87 


X 

0 

gTT 

N 

t 


f7T 

&r 

7T 

&7T 


b r 

|7T 

etc. 

3x 

0 


TV 

f7T 

27T 

far 

37r 

irr 

47T 

f7r 

5t 

etc. 

y = sin 3a; 

0 

1 

0 

- 1 

0 

1 

0 

- 1 

0 

1 

0 

etc. 


Example 2. Graph y — sin x + cos x by addition of ordinates. 

Solution. 1. Let yi = sin x and y 2 = cos x. Then, y = yi + y 2 . 

2. For any value of x, the ordinate y can be found geometrically, with compasses 
or dividers, by adding the corresponding ordinates on the graphs of y\ and y 2 , in 
Figure 67. 

3. If yi = 0, then y = y 2 ; this gives points A, B, and C; similarly we obtain 
D and E. If yi = — y 2 , then y = 0; this gives F and G. As many other points 
as desired can be obtained as described in Step 2; thus, to obtain H and K, 
double the ordinates of R and S, respectively. 



★EXERCISE 37 

Graph each function from x = 0 to x = 2rr: 


1. 

V 

= 

sin 2x. 3. 

y 

_ r 

5 cos x. 5. y = 

tan 2a;. 

7. y = cot 3x. 

2. 

y 

= 

cos 2x. 4. 

y 


1 sin x. 6. y = 

esc 3a;. 

8. y = sec 4a;. 

9. 

y 

= 

— 2 sin 3a;. 


11. 

y 

= sin 

12. y 

= COS 

10. 

y 

= 

— 3 cos 2a;. 



2 


O 

13. 

y 

= 

sin 2a; + cos x. 


16. 

y 

= X + COS X. 

19. y 

= cot X + cos X. 

14. 

y 


sin x — cos x. 


17. 

y 

= 2x + sin x. 

20. y 

= sec x + tan x. 

15. 

y 

= 

cos 2a; + sin x. 


18. 

y 

= tan x + sin x 

. 21. y 

= sin 2a; + esc x. 





























88 


TRIGONOMETRY 


REVIEW EXERCISE 38 

Change each angle to radian measure: 

1. 75°. 2. 15°. 3. 45°. 4. 126°. 5. 240°. 6. 315°. 7. 207°. 

The specified number is the radian measure for a certain angle. Change 
to degree measure, expressing the result correct to two decimal places: 

8. 3t r. 9. 5 tt/ 2. 10. 4. 11. 1. 12. - f. 13. - f. 

14. Construct an angle equal to 2 radians. 

15. In an isosceles triangle, one of the equal angles is 25°. Find all 
angles in radians. 

16. State in words the fundamental relation between a central angle in 
a circle, its radius, and the arc subtended by the angle. 

17. In a circle whose radius is 8 feet, what is the radian measure of a 
central angle which subtends an arc 15 feet long? 

18. How long an arc is subtended by a central angle of 125° in a circle 
whose radius is 15 inches? 

Without using a table, find the value of the function: 

19. sin fx. 20. cos fx. 21. tan f -it. 22. cot fx. 

Construct the angle in standard position on a coordinate system and find 
its functions from the figure, by direct use of Definition I, page 60: 

23. 2tt/ 3. 24. 5x/4. 25. tt. 26. - tt/ 2. 27. 7x/6. 

28. Sketch the graph of each of the trigonometric functions of 9 rapidly 
from memory, for 9 = 0° to 9 = 2x. 

29. Describe the variation of sin 6 as 9 varies from 0° to 180°. 

30. Describe the variation of tan 9 as 9 varies from 0° to 180°. 

31. Describe what is meant by writing sec 90° = oo. 

Express each function in terms of a function of 9 : 

32. sin (£x - 9). 33. cos (£x + 0). 34. tan (x - 9). 35. sec (-0). 

+Find each function by use of Table XIII; angles are expressed in radians: 

36. sin 1.03. 39. sin 1.034. 42. sin 1.324. 45. cos 2.10. 

37. cos .58. 40. cos .946. 43. cos 1.478. 46. sin 3.40. 

38. tan 1.46. 41. tan .563. 44. tan 1.94. 47. cos 3.60. 

Hint for Problem 44. Draw 1.94 radians in standard position on a coordinate 
system, (x — 1.94) = (3.14 — 1.94) = 1.20 radians. 

+Draw a graph of the function from x — — x to x = x: 

48. y = 4 sin 3x. 49. y = 2 sin ix. 50. y = sin 2x — cos 3x. 


CHAPTER VII 

THE FUNDAMENTAL IDENTITIES 

72. Identities and equations. An equation is a statement that 
its two sides or members are equal. An equation in which the two 
sides are equal for all permissible values of any variables involved is 
called an identical equation or for short an identity. An equation 
whose sides are not equal for all permissible values of the variables 
is called a conditional equation. Unless otherwise specified, the 
unqualified word equation refers to a conditional equation. A solution 
of an equation in an unknown x is a value of x which makes the two 
sides of the equation equal. 

Illustration 1. x — 3 = 0 is a conditional equation whose only solution is 
x = 3. The equality (o + 5) 2 = o 2 + 2 ab -f b 2 is an identity. 


73. The fundamental identities. If 6 is any angle, then 


(I) 


esc 0 - 


sin 0’ 


(II) 


sec 0 = 


1 

cos 0’ 


(III) 


cot 0 = 


1 

tan 0 


The reciprocal relations I, II, and III were proved on page 61 
by direct application of Definition I, page 60. In the same way 
we shall prove the following quotient relations IV and V, and the 
relations involving squares,* (VI), (VII), and (VIII). 

sin0 = tan 0 (y) cos0 = cQt e (yi) sin2 e + cog2 0 = 1 

v ' cos 0 sin 0 

(VII) tan 2 0 + 1 = sec 2 0. (VIII) 1 + cot 2 0 = esc 2 0. 

y 

Proof of (IV). ^4 = - = - • - = - = tan 0. [Using page 60] 

cos 9 x r x x 

r 

Proof of (VI). 1. From page 57, y 1 + z 2 = r 2 . (1) 

2. Divide by r 2 : V - 2 + ^ = 1; or, sin 2 9 + cos 2 0=1. 

Comment. To prove (VII) we would divide by x 2 in (1). 

* To indicate a power of a trigonometric function, we place the exponent between 
the function’s name and the angle. Thus, sin 3 9 means (sin 0) 3 . An exception is 
made in the case of the exponent — 1: we write, for instance, (sin 0) -1 and not 
sin -1 6 to mean 1/sin 9. 


89 





90 


TRIGONOMETRY 


Formulas I to VIII should be remembered in words. Thus, (IV) 
states that the tangent of an angle equals its sine divided by its cosine. 
Also, the identities should be recognized when slightly modified. 

Illustration 1. From (I), sin 9 = — 

esc 0 

From (IV), sin 9 = cos 9 tan 9. 

From (VI), sin 2 9 = 1- cos 2 9; cos 9 = ± Vl - sin 2 9. 

Example 1. Given that sin 9 = f and 9 is in quadrant II. Find all 

other functions of 9 algebraically. 

Solution. 1. From formula I, esc 9 = f. 

2. From (VI), cos 2 9=1- sin 2 0 = 1-*, or cos 2 9 = i f. Hence, 

cos 9 = ± = ± £. 

Since 9 is in quadrant II, only the minus sign applies: cos 9 = — %. 

3. From (II), sec 9 = - \. 

4. From (IV), tan 9 = ^ = - f; from (III), cot 9 = - *. 

Example 2. Given that cot 9 = * and 9 is in quadrant III. Find all 
other functions of 9. 

Solution. 1. From formula III, tan 9 = 

2. From (VII), sec 2 9 = 1 + W = W; sec 9 = = - V 1 - 

3. From (II), cos 9 = — *. 

4. From (IV), sin 9 = cos 9 tan 9 = — A • tt, or sin 9 = — ff. 

5. From (I), esc 9 = — \i. 

Comment. Problems of this nature were solved geometrically on page 64. 


EXERCISE 39 

Prove each identity by direct use of Definition I, page 60: 


1 . 


cos 9 


= cot 9. 


sin 9 
4. sin 9 esc 9 = 1. 


2. cos 9 = 


sin 9 


tan 9 

5. 1 + tan 2 9 = sec 2 9. 


3. sin 9 = 


cos 9 


cot 9 ' 

6. 1 + cot 2 9 = esc 2 9. 


By recalling a fundamental identity, write a proper right member involving 
only one function or one number: 

rj 1 _ ? - 1 _ „ 9. cos 9 sec 9 = ? 11. cos 9 tan 9 = 1 

cot 9 ' sec 9 10. tan 9 cot 9 = ? 12. sin 9 cot 9 = 1 

13. 1 - cos 2 9 = 1 15. esc 2 9 - cot 2 9 = 1 17. ± Vl - sin 2 9 = 1 

14. sec 2 9-1=1 16. sec 2 9 - tan 2 9 = 1 18. ± Vl + cot 2 9 = 1 









THE FUNDAMENTAL IDENTITIES 


91 


By use of Section 73, find all other functions of 0 if 0 is in the quadrant 
indicated by the Roman numeral. Do not introduce decimal values. 

19. sin 0 = *; (I). 21. cos 0 = - *f; (III). 23. tan 0 = *; (I). 

20. cos 6 = -h; (I). 22. sin 0 = - f; (IV). 24. cot 0 = f; (III). 

25. cot 0 = - V; (II). 28. esc 0 = - «; (III). 

26. tan 0 = — Y', (IV). 29. esc 0 - V; (II). 

27. sec 0 = - f; (II). 30. sec 0 = «; (IV). 

31. cos 0 = 1; (I). 33. tan 0 = 2; (III). 35. esc 0 = - 5; (IV). 

32. sin 0 = 1; (I). 34. cot 0 = - 3; (II). 36. sec 0 = - 2; (II). 

Compute the expression under the given condition: 

37. 3 tan 0(2 cot 0 + 1 - sec 2 0); if tan 0 = 2. 

38. 3 cot 2 0(sin 0 + esc 0); given sin 0 = 1. 

39. (tan 0 — 2) (sec 2 0 + 1); given cot 0 = 2. 

40. tan 0(sin 0 — 1); given cos 0 = f and 0 in quadrant IV. 


Express each function of x in terms of the given function: 

41. tan x. 42. sin x. 43. cos x. 44. cot x. 45. sec x. 46. esc x. 

Solution of Problem 44. 1. From formula III, cot x - —^— 

tan x 

2. From (VII), sec x = ± Vl tan 2 x- Hence, cos x = ± * - 

vi -f tan 2 x 

3. To complete the solution we would use sin x — cos x tan x, and then use 
formula I. 


Change each of the following expressions to a form involving no functions 
except sin x and cos x; combine into a single fraction and simplify: 


47. 

sec x cot x. 

49. 

cos x/sec x. 

51. 

cot x + tan x. 

48. 

sin x/ese x. 

50. 

cot x/sec x. 

52. 

tan x — sec x. 

53. 

cos x + sin x 

56. 

cot 2 X 

59. 

tan 2 x 

sec x -f- esc x 

1 + cot 2 X 

1 + tan 2 x 

54. 

CSC X 

57. 

1 

60. 

sec x — 1 

cot x + tan x 

tan x + cot x 

sec x + 1 

55. 

sec x 

58. 

tan x — cot x 

61. 

sec 2 x + esc 2 x 

cot x + tan x 

tan x + cot x 

sec x esc x 


sin x + tan x gg cot x + cos x > 

cot x esc x * 1 + sin x 

64 ___J_I_ 

sec x — tan x sec x + tan x 



















92 


TRIGONOMETRY 


74. Proofs of identities. The fundamental identities of Section 73 
can be used to prove other trigonometric identities. 

Rule 1. To prove an identity, alter one member until it assumes 
the same form as the other member, which is not altered. 


Note 1. Other methods will be illustrated, but generally Rule 1 is the most 
useful. Additional comments on identities are found in the Appendix, Note 5. 


Example 1. Prove the identity: tan 0 + 2 cot 0 
Solution. We alter the right member: 


sin 2 0 + 2 cos 2 0 
sin 0 cos 0 


tan 0 + 2 cot 0 


sin 2 0 + 2 cos 2 0 
sin 0 cos 0 


sin 2 0 2 cos 2 0 

sin 0 cos 0 sin 0 cos 0 

+ 2 = tan 0 + 2 cot 0. Q.E.D, 

cos 0 sin 0 


Example 2. Prove the identity: 


tan x + cot x = 


esc x 
cos x 


„ , , , , sin x . cos x sm 2 x + cos 2 x 

Solution. tan x + cot x = - — : - = —: - 

cos x sm x sm x cos x 


sm x cos x 


1 

sin x 


esc x 
cos x 


Q.E.D. 


CSC X 
COS X 


Optional solution. To prove an identity, we may alter both sides and finally 
exhibit them in identical form: 


tan x + cot x = 


sin x cos x 

cos x sin x 

sin 2 x + cos 2 x 
sin x cos x 


1 

sin x cos x 


1 

esc x _ sin x _ 1 

cos x cos x sin x cos x 


Example 3. Prove the identity: 


1 + cos x _ sin a; 
sin a; 1 — cos x 


Solution. On the right, multiply numerator and denominator by (1 + cos x): 

sin x 1 + cos x _ sinz(l + cosx) _ sinx(l + cos a;) _ 1 + cos x _ q ^ ^ 
1 — cos x 1 + cos x 1 — cos 2 x sin 2 x sin x 


★ Example 4. Prove the following identity, if a is acute: 

1 


[i 

Y 


sec a — tan a _ 
sec a + tan a sec a + tan a 


Solution. Rationalize the denominator on the left by multiplying both numer¬ 
ator and denominator under the radical by (sec a + tan a): 


V sec a — tan a _ 
sec a + tan a \ 


(sec a — tan a) (sec a + tan a) _ Vsec 2 a — tan 2 a 
(sec a + tan a) 2 sec a + tan a 


V (tan 2 a + 1) — tan 2 a. 

sec a + tan a 


sec a + tan a 


Q.E.D. 








































THE FUNDAMENTAL IDENTITIES 93 

Note 2. The following suggestions are useful in proving identities: 

a. If possible, avoid using formulas involving radicals. 

b. It may be convenient to express all functions in terms of sines and cosines. 

c. If one side involves only one function, it may be best to express everything on 
the other side in terms of this function. 

Note 3. If all functions on both sides of an identity are expressed in terms of a 
single function, the identity should appear obviously true after simplification. 
However, this method frequently is not convenient for proving the identity. 


EXERCISE 40 

Prove each identity by use of the fundamental identities; * leave one member 
unaltered unless otherwise directed j by the instructor: 

1 . cos ol esc ol = cot a . 5 . cos 2 d — sin 2 6 = 2 cos 2 6—1. 

2. sin 6 sec 6 = tan 6. 6. cos 2 6 — sin 2 0 = 1-2 sin 2 6. 

3. cot x + tan x = sec x esc x. 7. (sin a + cos a:) 2 = 1 + 2 sin a cos a. 

i. CSC 2 y tan 2 y = tan 2 y + 1. 8. (cot x + l) 2 = esc 2 x + 2 cot x. 

9. (esc x — cot a;) (esc X + cot x) = 1. 

10. (1 4- sin y)(l — sin y) = cos 2 y. 


11 . cot y = 


14. 

15. 

16. 

17. 

18. 
19. 


esc y 
sec y 

1 — cos 2 /3 


12. tan a = 


13. sin a = 


cos a 
cot a 


cos (3 


= sin /3 tan /3. 


tan 2 /3 cot 2 0 _ 

sec 2 (8 esc 2 (8 

tan 2 0 + 1 


cot 2 0+1 
cos 2 0 
1 — sin 0 
1 — tan 2 x 
1 + tan 2 x 
1 + esc x 


- tan 2 0. 

= 1 + sin 0. 

= 1—2 sin 2 x. 


20 . 

21 . 

22 . 

23. 

24. 


1 — cot 0 


= cos x + cot x. 25. 


esc a 
tan 0 — 1 
tan 0+1 1 + cot 0 

sin x — cos y _ sec y — esc x 
sin x + cos y 
sin y — cos y 
sin y + cos y 
sec x 

cot x + tan x 
sin x cos x _ 

2 sin 2 x — 1 tan x 
sin a + tan a 


sec y + esc x 
tan y — 1 
tan y + 1 

■ sin x. 


cot x 


sec x 

26. sec x + tan x = 


1 + sec a 
sin 2 x + sin x + cos 2 x 


= sin a. 


27. 


tan x — cos x cot x 


cos x 

sin x cos x 


esc x cot x sec x 

28. cot x — tan x = 2 cos x esc x — sec x esc x. 

* In case an ambiguous sign “±” arises, for simplicity assume that any angle in¬ 
volved is in quadrant I. 

f The instructor may permit the optional method of Example 2, Section 74. 


























94 


TRIGONOMETRY 


29. 


32. 


tan x — tan y _ cot y — cot x 
1 + tan x tan y cot x cot y + 1 


30. 


sec ' 1 x 


31. 


sin x — cos x 


tan x esc x — sec x cot x 


sec 2 x — 1 
= sin x cos x. 


sec* 3 + 2 tan x = x + tan 33 . 1 + s inl - 
1 + tan x 

34. (csc, + sec,)* = BeC,a: + 2ta,la: 


cos x 


sec x — tan x 


sir X 


35. 


1 


esc x — cot x esc x + cot x tan x 
36. cos 4 x - sin 4 x = 1 - 2 sin 2 x. 37. sec 4 x - tan 4 x = 
cos x cos y — sin x sin y _ cot x cot y —1 


1 + sin 2 x 


38. 

39. 


cos x sin y + sin x cos y cot x + cot y 

sin x cos y + cos x sin y _ tan x + tan y _ 
cos x cos y — sin x sin y 


1 — tan x tan y 

•kProve each identity without altering the right side: 


40. 


41. 


1 + sin x 
cos x 

1 

sec x + tan x 


1 — sin x 
1 — sin x 


43 


44. 


• 4 


+ sin x 


= sec x — tan x. 


V csc x 
esc x 

42. 1=3 = - -S5J!-. 45. 

\ sec x + 1 1 + cos x > sec x 


— cot x 1 — cos x 


+ cot x 


— tan x 


sm x 
cos x 


+ tan x 1 + sin x 


75. Trigonometric equations in one unknown. On account of 
the periodicity of the trigonometric functions, any trigonometric 
equation of the types we shall consider usually has infinitely many 
solutions. In this book, unless otherwise directed, to solve a trigo¬ 
nometric equation will mean to find only those solutions which are 
'positive or zero and less than 360°. 

Note 1. Recall from Section 53, or Section 59, that the functions of (180° ± a ) 
and of (360° — a) are numerically equal to the functions of a. 

A trigonometric equation of the most simple type in a single un¬ 
known angle 6 is an equation which is linear in one function of 6 
and involves 6 in no other way. The equation immediately gives us 
the value of this function of 6. Then, there are usually two cor¬ 
responding values of 6 between 0° and 360°. We saw geometrically 
why this fact is true when sin 6 is given, in Example 2, page 64. 
































THE FUNDAMENTAL IDENTITIES 95 

Illustration 1. The following equation is linear in cos 0; that is, cos 0 is in¬ 
volved only to the first power. 

Example 1. Solve: 2 cos 0 — 1 = 0. 

Solution. 1. 2 cos 0 = 1; hence, cos 0 = £. 

2. Recall that cos 60° = ^ and that the cosine is positive in quadrants I and IV. 
Hence, cos 0 = \ if 0 = 60° or 0 = 360° — 60°. The positive solutions less than 
360° are 60° and 300°. 

Comment. Other solutions are — 60°, 420°, 660°, etc., which are obtained by 
adding integral multiples of 360 3 to 60° and 300°. For every integral value of n, 
each of the following expressions is a solution: 

0 = 60° + n(360°); 0 = 300° + n(360°). 

Example 2. Solve: cos 0 = —.9872. 

Solution. 1. Let a be the acute angle such that 
cos a = .9872. From Table VII, a = 9° 10'. 

2. The cosine is negative in quadrants II and III. 

Hence, we see from Figure 68 that 

0 = 180° - 9° 10' = 170° 50' and 0 = 189° 10'. 

A trigonometric equation in one unknown d, which is in the 
quadratic form in one function of 6, can be solved by use of methods 
met in dealing with quadratic equations in algebra. 

Illustration 2. The equation in Example 3 below is in the quadratic form in 
sin x because, if we place v = sin x, the equation becomes 2a 2 — v — 1 = 0. 

* Example 3. Solve: 2 sin 2 x — sin x — 1 = 0. 

Solution. 1. Factor: (2 sin x + l)(sin x — 1) =0. The equation is 

satisfied if 2 sin x + 1 = 0, or if sin x — 1 = 0. 

2. Solve 2 sin x + 1 = 0: sin x = — £; x = 210° and x = 330°. 

3. Solve sin x — 1 =0: sin x = 1; x = 90°. 

Comment. We would say that the given equation in Example 3 is equivalent 
to the equations 2 sin x + 1 = 0 and sin x — 1 = 0 because the solutions of 
these gave us all solutions of the original equation. 

To solve an equation by use of factoring, the equation must first 
he written with one member zero. 

Example 4. Solve: esc x tan x = tan x. 

Solution. 1. Subtract tan x from both sides; then factor: 

esc x tan x — tan x = 0; tan x(csc x — 1) = 0. 

2. If tan x = 0, then x = 0° or x = 180°. If esc x — l — 0, then x = 90°. 

Incorrect solution. Divide both sides by tan x: esc x = 1; x = 90°. 

Comment. If both sides of an equation are divided by an expression involving 
the unknowns, solutions may be lost. Thus, the division by tan x caused the 
loss of the solutions 0° and 180°, for which tan x = 0. 





96 


TRIGONOMETRY 


EXERCISE 41 

Solve each equation, using Table VII if necessary: 


1. 

tan x 

— 

1. 



10. 

3 cot 

X + 

V3 = 

0. 

19. 

tan x = 

1.621. 


2. 

cos X 

= 

1 

2 * 



11. 

2 cos 

x — 

V2 = 

0. 

20. 

tan x = 

4.449. 


3. 

sec x 

= 

1. 



12. 

3 tan 

x — 

V3 = 

0. 

21. 

cos x = 

- .8225. 

4. 

sin x 

= 

0. 



13. 

cos X 

_ 5 
“ 2 • 



22. 

cot x = 

- 7.770. 

5. 

sin x 

= 

1 

2 < 



14. 

sin x 

= 3. 



23. 

sin 2 x = 

1 

4- 


6. 

CSC X 

= 

- V2. 



15. 

sec x 

_ 1 

— 2 • 



24. 

tan 2 x = 

1. 


7. 

tan x 


V 3 . 



16. 

CSC X 

_ 1 

— 



25. 

sec 2 x = 

= 2. 


8. 

cot X 

= 

- 1. 



17. 

sin x 

= .4823. 


26. 

cot 2 x — 

= 3. 


9. 

2 sin x + V2 = 

0. 


18. 

sin x 

= .8526. 


27. 

4 sin 2 x 

- 3 = 

0. 

28. 

3 tan 2 

X 

- 1 = c 

I. 




35. 

(cot X 


1)(3 

cot 3 : + V3) = 

0. 

29. 

CSC 2 X 

— 

2 = 0. 





36. 

sin 2 x 

- 

2 sin 

x + 1 = 

0. 


30. 

(sin x 

+ 

1)(2 sin 

X 


• 1) 

= 0. 

37. 

CSC 2 X 

- 

4 esc 

x + 4 = 

0. 


31. 

(tan x 


V 3) (tan x 

+ D 

= 0. 

38. 

(sec 2 x - 

• 4) (2 

sec x + 

1) =0. 


32. 

2 cos 2 

X 

— cos X 

— 

1 

= 0. 

39. 

(2 cos 

2 X 

— 1)(4 esc 2 x - 

- 1) - 

0. 

33. 

sec 2 x 

— 

sec x — 

2 

= 

0. 


40. 

sec 2 x 

- 

3 sec 

x + 2 = 

0. 


34. 

(esc X 

- 

l)(csc X 

+ 

V2) 

= 0. 

41. 

2 cos 2 

X 

-f cos 

x — 1 = 

0. 


42. 

cot 2 X 

= 

cot X. 


43. 

sin 2 x 

= sin x. 


44. 2 

COS 2 X = 

— cos 

X. 

45. 

2 sin 2 

X 

+ V2 sin 

X 

= c 

). 

48. 

sin x 

sec 

: 2 X — 

2 sin x = 

= 0. 


46. 

3 tan 2 

X 

+ V3 tan 

X 

= 0. 

49. 

cos X 

cot 2 x — 

3 cos x = 

= 0. 


47. 

2 esc 2 

X 

+ CSC X 

= 

0. 



50. 

2 tan 

x sin x - 

- tan x = 

= 0. 



SUPPLEMENTARY PROBLEMS 

51. 2 sin 4 x - 3 sin 2 x + 1 = 0. 54. 3 tan 3 x - tan * = 0. 

52. 3 cot 4 x + 2 cot 2 x - 1 = 0. 55. 4 cos 3 x = 3 cos x. 

53. 2 sin 3 x - sin x = 0. 56. 6 cot 2 x - 5 cot x + 1 = 0. 


57. 2 sin 2 x — 3 sin x — 1 = 0. 

Hint for Problem 57. 1. Let v = sin x; then 2v 2 - Sv - 1 = 0. By use of 

3 ± V9 + 8 


the quadratic formula we find 


v = sin x = 


2. Compute the decimal values of sin x by use of Table I. Then use Table VII. 


58. sin 2 x — 8 sin x + 15 = 0. 

59. 3 cot 2 x — 5 cot x — 1 = 0. 

60. 2 sec 2 x — 3 sec x — 8 = 0. 

64. -3-(- 3 sin x — 1 = 0. 

sin x 


61. 3 sin 2 x + 3 sin x = 8. 

62. 3 tan 2 x — tan x — 3. 

63. 3 esc 2 x + 5 esc x = 3. 

65. 5 — 3 cos x = 


cos X 






THE FUNDAMENTAL IDENTITIES 


97 


76. Equations solved by the aid of identities.* In solving a trigo¬ 
nometric equation in x, our aim is to find one or more equations, 
each involving only one function of x, which are equivalent to the 
given equation. To accomplish this aim, it may first be useful to 
substitute identical expressions for the functions in the given equa¬ 
tion, in one of the following ways, by use of fundamental identities. 

A. Express each function of x in terms of one function of x. 

B. Express each function of x in terms of sin x and cos x. 

Example 1. Solve: tan 2 x + 3 sec x + 3 = 0. 

Solution. 1. Use tan 2 x = sec 2 x — 1: sec 2 x + 3 sec x + 2 = 0. 

2. Factor: (sec x + 2)(sec x + 1) = 0; sec x = — 2 or sec x = — 1. 

3. The solutions are 120°, 180°, and 240°. 

Example 2. Solve: sin 0 + cos 0 = 0. 

Solution. 1. Divide both members by cos 0: tan 0 + 1=0. 

2. tan 0 = - 1; hence, 0 = 135° or 0 = 315°. 

Comment. If cos 0 = 0, then 0 = 90° or 0 = 270°. By substitution we see 
that neither value is a solution of the given equation. Hence, no solutions were 
lost through dividing both sides by cos 0. 

77. Extraneous solutions. In solving an equation, it is some¬ 
times useful to employ one or both of the following operations: 

A. Square both sides (or raise both to any specified power). 

B. Multiply both sides by an expression involving unknowns. 

The new equation obtained by use of (A) or ( B ) may have solutions 
not possessed by the given equation; such solutions are called ex¬ 
traneous solutions. Hence, if (A) or ( B ) is used, all values obtained 
for the unknowns must be tested in the original equation to reject 
extraneous solutions if any. 

Example 1. Solve: cos x + 1 = sin x. (1) 

Solution. 1. Use sin x = =t Vl — cos 2 x: cos x + 1 = ± Vl — cos 2 x. 

2. Square both sides: (1 + cos x) 2 = 1 — cos 2 x; 

1+2 cos x + cos 2 x — 1 — cos 2 x; 2 cos x + 2 cos 2 x = 0; 

2 cos x(l + cos x) = 0. (2) 

From (2), x = 90°, x = 180°, or x = 270°; these may be solutions. 

3. Test all values of x by substituting in (1). Test x = 270°. 

Does cos 270° + 1 = sin 270°? Does 0 + 1 = - 1? No. 

Hence, 270° is not a solution. Similarly, we find that 90° and 180° are solutions. 

* In a brief course, the instructor may desire to omit the remainder of this chapter. 




98 


TRIGONOMETRY 


2 cos 0 — 


= 0 . 


Example 2. Solve: sec 0 — 2 cos 6 tan 9 0. 

Solution. 1. Express each function in terms of sin 6 and cos 6: 

1 

cos 6 

2. Multiply both sides by cos 6: 

Use cos 2 0 = 1 — sin 2 0: 

Hence, 


(3) 


cos 0 

1 — 2 cos 2 0 — sin 0 = 0. 

1 — 2(1 — sin 2 0) — sin 0 = 0. 

2 sin 2 0 — sin 0 — 1 =0. 

3. Factor: (2 sin 0 + l)(sin 0 - 1) = 0; sin 0 = 1 or sin 0 = - b 

4. From Step 3, 0 = 90°, 0 = 210°, and 0 = 330 . 

5. Test. By substitution in (3), we find that 210° and 330° are solutions. If 

0 = 90°, the original equation has no meaning because sec 90° = <*> and 
tan 90° = . Hence, 90° is not a solution. 

Note 1. Frequently, it is impossible to tell by mere inspection whether a 
specified equality is an identity or an equation. If the equality happens to be an 
identity, and if we proceed as we would in solving an equation, then eventually 
we should obtain one or more obvious identities from the given equality 


EXERCISE 42* 

Solve each equation; use Table VII if necessary 

1. 2 cos 2 x + sin x — 1 = 0. 

2 . 3 + 3 cos x = 2 sin 2 x. 

3. cos 2 x + sin x + 1 = 0. 

4. 3 sin 2 x — cos 2 x — 1 = 0 

5. 2 sin 2 x — 2 cos 2 x = 3. 

6. V3 cot x + 1 = esc 2 x. 


7. sec 2 x — 1 = tan x. 

8. 2 sec x + 3 = 2 cos x. 

9. 3 esc x + 2 = sin x. 

10. esc 2 x — cot £ — 1 = 0. 

11. tan 2 x + sec 2 x = 9. 


12. cot 2 x + esc 2 x = 19. 

13. tan £ = 3 cot x. 15. 3 sin 0 = V3 cos 0. 17. tan 2 £ - sec £ = 1. 

14. cos x — sin x. 16. 3 tan 0 — cot 0 = 0. 18. 3 cos 6 — sin 0. 


19. 

cot 2 X + 3 CSC £ ■ 

|3-0. 

25. 

sin 

X 

+ 

1 = 

cos 

£. 

20 . 

CSC 

x — 2 sin £ = 

= cot X. 

26. 

sin 

X 

= 

1 + 

cos 

X. 

21. 

sec 

x = cos £ — 

tan x. 

27. 

cos 

X 

- 

1 - 

sin 

X. 

22 . 

3 sec £ + 3 tan x 

= 2 cos x. 

28. 

cot 

X 

+ 

1 = 

CSC 

X. 

23. 

CSC 

£ + cot x = 

2 sin x. 

29. 

tan 

X 

= 

sec 

£ + 

1. 

24. 

CSC 

x = sin £ — i 

cot X. 

30. 

sec 

X 

+ 

tan 

£ = 

1. 



31. sin 2 £ sec x + 2 sec 

x — 

cos 

X 


3 tan x. 




32. 3 cot £ 

— COS 2 X CSC 

x — 

2 esc 

./■ 

+ sin x - 

= 0 

33. 9 

CSC 2 

x = 4 tan 2 x. 

34. 2 cot 2 

X = 

sec 2 

: X 


35. 

sec 2 

£ - 

36. 

see 

2 £ + 3 cot 2 X 

= 5. 

38. 

sec 

X 

+ 

3 = 

tan 

X. 

CO 

3 esc 2 £ — tan 2 x 

= 1. 

39. 

cot 

X 

= 

CSC 

£ + 

2. 


* Avoid introducing radicals if possible. 



THE FUNDAMENTAL IDENTITIES 


99 


REVIEW EXERCISE 43 

Prove each identity by direct use of equations 1, page 60: 

, n esc 0 n cot a 0 tan 2 0 , cot 2 0 1 

I. cot 0 = - v 2. cos a =-3. —^ H- 5-5 = 1. 

sec 0 esc a sec 2 0 esc 2 0 

By use of the fundamental identities, find all other functions of 6 if d is 
in the quadrant indicated by the Roman numeral. Do not introduce decimal 
values in case surds occur. 

4. sin 0 = A; 0 in (I). 7. cos 0 = — ^ 5 -; 0 in (II). 

6 . tan 0 = 0 in (III). 8. sec 0 = 3; 0 in (IV). 

6 . cot 0 = ¥; 0 in (HI). 9. esc 0 = 2 ; 0 in (II). 

10 . Express sin 0 in terms of cot 0. 

II . Express esc 0 in terms of tan 0. 

Solve each equation; vke Table VII if necessary: 


12 . 

cot 0 

= 1 . 



16. 

CSC 

0 = 

2. 


20 . 

tan 0 

= . 

7173. 

13. 

sin 0 

_ 1 
— 



17. 

tan 

0 = 

-Vs 


21 . 

sin 0 

= 

- .2812. 

14. 

cos 0 

= — 

h 


18. 

sin 

0 = 

.3800. 


22 . 

cos 0 

= 

- .9580. 

15. 

sec 0 

= - 

V2 


19. 

sin 

0 = 

.5519. 


23. 

sin 0 

= 

- .2518 

24. 

1 + sin x ■ 

- 2 

sin 2 

x = 

= 0. 


30. 

cot X = 

3 tan 

X. 


25. 

sec 2 x 

- 4 

sec ; 

* + 

4 = 

= 0. 


31. 

2 

sin 2 x 

+ sin 

X = 

= 0 . 

26. 

3 cot 2 

* + 

V 3 

cot 

x = 

= 0. 


32. 

3 

tan x 

- 1 = 

0. 


27. 

2 cot 

x sin 

x — 

cot 

, X ■ 

- 0 . 


33. 

2 

sin x - 

- cosx 

= 

0. 

28. 

3 cos 2 

x — 

sin 2 

x — 

■ 1 

= 0. 


34. 

5 

sin x 

+ 2 = 

0. 


29. 

sin 2 x 

+ cos X 

+1 

= 

0. 


35. 

3 

cos X 

+ 4 = 

0. 



Prove each identity by use of the fundamental identities: 


36. 

37. 

38. 

39. 


sin x 

sec x 

40. 

sin x 

cos - 

sin x + cos x 

sec x -j- esc x 

sin x — tan x 

cos x — 1 

sin x 

cos X 

41. 

cos X 

1 

sin x + sec x 

~ cos x -j- CSC X 

cos x — sin x 

~ 1 — tan x 

sin x — sec x 

COS X — CSC X 

42. 

sin x 

1 

sin x + sec x 

COS X + CSC X 

sin x — cos x 

1 — cot a: 

cos X 

cot X 

CO 

sin x 

tan x 

cos x — sin x 

cot x — 1 

sin x — cos x 

tan x — 1 


44. 


tan 0 — esc 0 sin 2 0 — cos 0 


tan 0 + esc 0 
esc 0 


sin 2 0 + cos 0 
cos 0 


esc 0 — tan 0 cos 0 - sin 2 0 


46. 

























100 


TRIGONOMETRY 


Prove each identity by use of the fundamental identities: 


2 sin a 


46. cos 1 « tan a = 8ec „ + cos a ^ a Bec a 

47 . sec 4 a + tan 4 a = 1 + 2 sec 2 a tan 2 a. 

48. esc 4 a + cot 4 a = 1 + 2 esc 2 a cot 2 a. 
sin a + cot a 


49. 

50. 


cos a sin a 
1 


— 1 = cot 2 a + sec a. 
sin a 


cot a — tan a cos a — sin 2 a sec a 


sin x — sec y _ cos y — esc x 

sin x + sec y ~ cos y + esc x 

sin x _ cos y 
sin x + sec y cos y + esc x 

53. = ( S ec x - esc x)(l + lin x cos x). 

esc x sec x 

sin a cos + cos a sin ft _ tan a cot ft + 1 
cos ol cos j8 — sin a sin /3 cot — tan a 

_ sin a _ ___ 

sec a + 2 sin a — 2 sin 2 a sec a cot a — tan a + 2 

sec oi esc ol — 2 tan a _ cos a — sin a 

° 6 ' cos a + sin a sin a cos a 


SUPPLEMENTARY PROBLEMS 

Determine whether the equality is an identity or a conditional equation. 
If not an identity, solve for the unknown: 

57. 3 tan x - 3 = - V3 cot x. 

58. 8 sin 4 x + 10 cos 2 x — 7 = 0. 

59. cot x + tan x esc x — tan x = 0. 

60. 2 sin a tan a — 2 sin a + tan a = 1. 

61. (sin x - cos x) 2 = 1 - 2 cot x sin 2 x. 

62. (1 + sec x)(l - cos x) = tan x sin x. 

63. (sin x — cos x) 3 = 3 sin x - 2 sin 3 x + 2 cos 3 x — 3 cos x. 

64. 3 — cos 2 x esc x — 3 sin 2 x — 2 cos 2 x. 

65. 2 cos a cot 2 a — 6 cos a — cot 2 a = — 3. 

66. (tan x + cot x) 3 = tan x sec 2 x + cot x esc 2 x + 2 tan x + 2 cot x. 

67. sin a tan a — 2 sin 2 a cos a + tan a — 2 tan a cos 2 a = 0. 

68. 5 tan 2 a sin a — sin a sec 2 a + sec 2 a esc a = 5 tan 2 a esc a. 


















CHAPTER VIII 

ADDITION FORMULAS AND RELATED TOPICS 


78. The sine and cosine of the sum of two angles. Let 

represent any two angles. Then, it can be proved that 

sin (a + /3) = sin a cos (i + cos a sin /3; 

cos (a + 0) = cos a cos /3 — sin a sin /S. 

Illustration 1. By use of (II), 

cos (30° + 60°) = cos 30° cos 60° — sin 30° sin 60° 

V3 1 _ 1 V3 

2 2 2 2 U ' 

To check this result we notice that cos (30° + 60°) = cos 90° = 0. 

Illustration 2. By use of page 70 we could verify that (I) and (II) are correct 
if a or /3 is any integral multiple of 90°. Thus, if a = 90°, (II) gives 

cos (90° + /3) = cos 90° cos /3 — sin 90° sin /3 = — sin /3, 
which agrees with equations 1, Section 57, on page 69. 

All formulas in this chapter should be remembered in words as well 
as in symbols. Thus, (I) states that the sine of the sum of two angles 
equals the sine of one of the angles times the cosine of the other plus 
the cosine of the first angle times the sine of the other. We call (I) the 

addition formula for the sine and (II) the addition formula for the 
cosine. 

Proof of (I) and (II) when a and (3 are positive and acute. 

1. Place a in its standard position on a coordinate system, as described 
on page 59. Place /3 with its vertex at the origin and its initial side OP 
on the terminal side of a. Then, in Figure 69 or Figure 70 on page 102, 
angle MOQ is (a + /3) and this angle is in its standard position. 

Comment. Figure 69 applies if a + (3 < 90°, and Figure 70 if a + /3 > 90°. 
The proof is worded to apply to either figure with the understanding that any 
line segment parallel to a coordinate axis is positive or negative according as its 
direction is the same as or opposite to the positive direction on the axis. Thus, 
OL is positive in Figure 69 and negative in Figure 70. In a first reading of the 
proof, think only of Figure 69. 


a and /3 


(I) 

(II) 


101 


102 


TRIGONOMETRY 


2. From any point Q on the terminal side of /3, drop a perpendicular to 
the terminal side of a. Draw QL and PM perpendicular to OX, and PH 
parallel to OX. 



Fig. 69 



3. Angle HQP equals a because* HQ and QP are respectively per¬ 
pendicular to the sides OM and OP of a. 

4. Triangles HQP, OMP, and OPQ are right triangles. Hence, 

from triangle OMP, MP = OP sin a; OM — OP cos a; (1) 

from triangle HPQ, HQ = PQ cos a; HP = PQ sin a; (2) 

OP PO 

from triangle OPQ, ^ = cos (8; ^ = sin /S. (3) 

6 . The coordinates and radius vector of Q are x = OL, y = LQ, 
and r = OQ. Therefore, from Definition I, page 60, 


sin (a + /3) 
sin (a + /3) 


LQ _ LH + HQ MP HQ 
OQ OQ OQ + OQ’ 
OP sin a PQ cos a 
OQ + OQ 


(Since MP = LH) 
[Using (1) and (2)] 


Hence, sin (a + (3) = sin a cos /3 + cos a sin |8. [Using (3)] 


6. From Definition I, page 60, cos (a + /3) = 

a x ^ - OM HP OP cos a PQ sin a 

3 ( “ + P) - OQ ~ W = OQ - OQ ~' 1 

cos (a + /3) = cos a cos /3 — sin a sin /3. 


OL OM - LM 
OQ OQ ’ 

[Using (1) and (2)] 

[Using (3)] 


* The following theorem is proved in plane geometry: if the sides of one acute angle 
are respectively perpendicular to the sides of another acute angle , the angles are equal. 


















ADDITION FORMULAS AND RELATED TOPICS 


103 


★Note 1. The preceding method could be used to prove (I) and (II) under 
any * sufficiently explicit assumptions about the values of a and 0. For any special 
case, the figure would be constructed exactly 

as stated in Steps 1 and 2 of the preceding v 

proof. Thus, Figure 71 would apply if 
90° < a < 180°, 0 >0°, and a + 0 < 180°. 

In Figure 71, 6 — 180° — a and HQ is nega¬ 
tive, but nevertheless Steps 4, 5, and 6 of 
the preceding proof hold without alteration. 

Without further discussion we shall 
proceed under the assumption that 
(I) and (II) are known to be true for 
all values of a and 0, positive, nega¬ 
tive, or zero. 

Example 1. Find sin 105° by expressing 105° as a sum. 

Solution, sin 105° = sin (60° + 45°) = sin 60° cos 45° + cos 60° sin 45°; 

V2 _ a/6 + V2 2.449 + 1-414 _ 



-mo.-:*:£+§ 


= .966. 


79. Addition formulas for the tangent and cotangent. 


tan (a + 0) 


tan a + tan 0 
1 — tan a tan 0 


cot (a + 0) = 


cot a cot 0 — 1 
cot a + cot 0 


(HI) 

(IV) 


Proof of (III). 1 . tan (a + 0) 


sin (a + 0) _ 
cos (a + 0)’ 


tan (a + 0) 


sin a cos 0 + cos a sin 0 
cos a cos 0 — sin a sin 0 


[Using (I) and (II)] 


2. Divide both numerator and denominator by cos a cos 0: 

sin a cos 0 cos a sin 0 sin a sin 0 

. . _ cos a cos 0 cos a cos 0 _ cos a cos 0 . 

an (a + 0) a cog p sin a sin 0 ^ _ sin a _ sin 0 ’ 

cos a cos 0 cos a cos 0 cos a cos 0 

the final fraction is identical with the right side in (III). 

To prove (IV), we would proceed as for (III), except that we 
would divide the corresponding numerator and denominator by 
sin a sin 0. 

* A non-geometrical proof that (I) and (II) hold for all angles a and 0 is given in 
the Appendix, Note 6, with the preceding proof as a starting point. 























104 


TRIGONOMETRY 


Illustration 1. cot 75° — cot (45° + 30°) = 
V3 - 1 


cot 45° cot 30° — 1 


cot 75° = 


cot 45° + cot 30° ’ 
(Vjj - 1)<V3 - 1) _ 4 - 2V3 


1 + V 3 (V3 + i)(V3 - 1; 


= 2 - V 3 . 


Note 1. We do not introduce addition formulas for the secant and cosecant 
because these formulas would lack simplicity and, moreover, would have rela¬ 
tively few applications. In this chapter, when referring to the functions of an angle, 
unless otherwise specified we shall mean only its sine, cosine, tangent, and cotangent. 


80. Functions of the difference of two angles. 

Illustration 1. To find sin (60° - 45°) we may use formula I, page 101, with 
a = 60° and 0 = — 45°: 

sin (60° - 45°) = sin [60° + (- 45°)] 

= sin 60° cos (— 45°) + cos 60° sin (— 45°) 

= sin 60° cos 45° - cos 60° sin 45°. [Using Section 55, page 68 ] 


Instead of using (I) to (IV) as in Illustration 1 when finding 
functions of the difference of two angles, for convenience we derive 
the following new formulas for solving such problems. 


sin (a — 0 ) 
cos (a — 0 ) 

tan (a — 0 ) 
cot (a — 0 ) 


sin a cos 0 — cos a sin 0 . 

(V) 

cos a cos 0 + sin a sin 0 . 

(VI) 

tan a — tan 0 

1 + tan a tan 0 

(VII) 

cot a cot 0 + 1 
cot 0 — cot a 

(VIII) 


sin (— 0) = — sin 0. 


Proof of (VI). 1. To obtain cos (a - 0), substitute (- 0) for 0 in (II): 

cos (a — 0) = cos [a + (- /3)] = cos a cos (- 13) - sin a sin (- 0). (1) 

2. From page 68, cos (— 0) = cos (3 

3 . Hence, from (1), 

cos {a - 0) = cos a cos 0 + sin a sin (3. 

Illustration 2. sin 75° = sin (135° — 60°) 

= sin 135° cos 60° — cos 135° sin 60° 


V2 

2 


V2 

~2~ 


V3 

2 


V2 + V6 



★Note 1. A geometric method like that 
in Section 78 can be used to prove special 
cases of (V) and (VI): Figure 72 would apply 
if a and (3 are positive and acute, and if p IG . 72 

a > (3. The figure was constructed by 

Steps 1 and 2 of the proof on page 101, with (— /3) substituted for (3. 















ADDITION FORMULAS AND RELATED TOPICS 


105 


EXERCISE 44 

Solve each problem by use of formulas I to VIII: 

1. Find cos 135° and cot 135° by using functions of 90° and 45°. 

2. Find sin 225° and tan 225° by using functions of 180° and 45°. 

3. Find cot 330° and cos 330° by using functions of 270° and 60°. 

4. Find tan 210° and sin 210° by using functions of 180° and 30°. 

5. Find sin 315° and tan 315° by using functions of 360° and 45°. 

6. Find cos 150° and tan 150° by using functions of 180° and 30°. 

7. Find sin 225° and cot 225° by using functions of 270° and 45°. 

8. Find cos 45° and tan 45° by using functions of 180° and 135°. 

Find the functions of each angle without using tables; any result may be 
left in radical form. Solve first by expressing the angle as a sum , and, second, 
as a difference of convenient angles like 30°, 45°, 60°, 120°, 135°, etc. 

9. 75°. 10. 105°. 11. 165°. 12. 195°. 13. 285°. 14. 255°. 


Expand by use of formulas I to VIII and insert known values: 


15. sin (30° + a). 

16. cos (45° - a). 

17. tan (i8 + 45°). 

18. cot (45° - A). 


19. cos (A — 45°). 

20. sin (60° - A). 

21. cot (a - 60°). 

22. tan (a + 135°). 


23. sin (270° + a). 

24. cos (270° - a). 

25. tan (x — a). 

26. cot (Fx + a). 


27. State each of formulas I to VIII in words. 

28. (a) Compute sin 30° + sin 45°. (6) Compute sin (30° + 45°). 

29. (a) Compute tan 45° + tan 60°. (6) Compute tan (45° + 60°). 


Do not expand or substitute values of functions on the left side; complete 
each equality with a right member which is an explicit number or a single 
function of some angle. 

30. sin 25° cos 65° + cos 25° sin 65° = ? 

31. cos 125° cos 55° - sin 125° sin 55° = ? 

tan 15° + tan 30° ? cot 10° cot 50° - 1 ? 

6 ' 1 - tan 15° tan 30° ’ ^ cot 10° + cot 50° 

cot 25° cot 110° + 1 0 tan 240° - tan 15° _ ? 

6i ' cot 110° - cot 25° ' 1 + tan 240° tan 15° C 

36. sin A cos 10° — cos A sin 10° = ? 

37. cos A cos 40° + sin A sin 40° = ? 

38. cos 15° cos 60° - sin 15° sin 60° = ? 

39. cos 50° sin A — sin 50° cos A = ? 

40. sin 2 A cos A + cos 2 A sin A = ? 

41. cos 3 B cos B + sin SB sin B = ? 






106 


TRIGONOMETRY 


SUPPLEMENTARY PROBLEMS 


42. Prove formula IV. 44. Prove formula VII. 

43 . Prove formula V. 45. Prove formula VIII. 


Prove each of the following identities: 

46. sin (30° - x) + cos (x - 120 °) = 0 . 

47. sin (45° + x) - sin (45° - x) = V2 sin x. 

48. cos (30° + x) cos (30° - x) - sin (30° + x) sin (30° - x) = f. 


49. cos (x + x) cos (x — x) — sin (x + x) sin (x — x) — 1 . 


50. cot (45° + x) = 


1 — tan x 
tan x + 1 


51. tan (45° - x) = 


cot x — 1 
cot X + 1 


52. Draw a special case of Figure 69, differently lettered, and prove 
(I) and (II) by use of your figure. 

53. Repeat Problem 52 by use of a figure like Figure 70. 


Without tables, find the functions of ( a + /3) and of (a — (3) under the 
given conditions, if a and f3 are positive and acute: 

54. sin ol — if; cos /3 = 57. tan a = f; cot ft = 

55 . tan a = sin p = If. 58. sin a = tV; cot /3 = f. 

56. sin a = A; cos |3 = ff. 59. sin a = tan /3 = hr- 


Without tables, find the quadrant in which (a - fi) lies under the given 
conditions, if a and (3 are positive and acute: 

60. tan a = 3; cot j3 = f. 62. sin a = A; cos /3 = 

61. tan a = f; cot 0 = 3. 63. sin a = f; cos 0 = ^r. 

64. Express sin (a + /3 + 7 ) and cos (a + (3 + y) in terms of the sines 

and cosines of a, 1 3, and 7 . 

Hint. sin (a + (3 + 7 ) = sin [(a + /3) + 71- 


tan a. 

tan y + tan (x + z) 

1 — tan y tan (x + z) 

Express in terms of tan A and tan B, without using radicals: 

3 sin A cos B — cos A sin B _ c os A sin B + 3 cos A cos B 
67, 3 sin A sin B + cos A cos B' ' sin A sin B + 3 cos A sin B 

69. Prove (V) and (VI) by use of Figure 72. 


Prove each identity: 

tan (a + j3) - tan j3 
1 + tan (a + (3) tan (3 
6 _ tan x + tan (y + z) 

1 — tan x tan (y + z) 









ADDITION FORMULAS AND RELATED TOPICS 107 

81. Double angle formulas. On substituting |8 = a in formulas 
I to IV, we obtain the following formulas for the functions of 2a, 
where a is any angle. 


sin 

2a 

= 2 sin a cos a. 

(IX) 

cos 

2a 

= cos 2 a — sin 2 a\ 

(X a ) 

cos 

2a 

= 2 cos 2 a — 1; 

(X 6 ) 

cos 

2 a 

= 1 — 2 sin 2 a. 

(x e > 

tan 

2 a 

2 tan a 

1 — tan 2 a 

(XI) 

cot 

2a 

_ cot 2 a — 1 

2 cot a 

(XII) 


Proof of (X). To obtain cos 2a, place /3 = a in (II): 

cos 2a = cos (a + a) = cos a cos a — sin a sin a = cos 2 a — sin 2 a, 

which proves (X„). To prove X*, substitute sin 2 a = 1 - cos 2 a in (X a ). 
To prove (X c ), substitute cos 2 a = 1 — sin 2 a in (X a ). 

Illustration 1. By use of (X„) with a = 60°, we obtain 

cos 120° = cos 2(60°) = 2 cos 2 60° - 1 = 2(*) 2 - 1 = - 
From (Kb) and (X c ) we obtain the following useful formulas: 


2 cos 2 a = 1 + cos 2 a ; 

2 sin 2 a = 1 — cos 2a. 

(XIII) 

82. Half-angle formulas. By use of formulas XIII, 
prove that, for all values of the angle 6, 

we shall 

. 0 1 1 — cos 6 

Sm 2 = 2 ’• 

0 /1 + cos 0 

C0S 2 = * V 2 ; 

(XIV) 

,0 /1 — cos 0 

t “2- ± V 1 + cos9 ’ 

, 0 /I + COS 0 

2 = ± V 1 — CO s $ 

(XV) 

Proof of (XIV). In (XIII), a is one half of 2a. Hence, (XIII) 
the square of the sine or cosine of one half of an angle in terms of 
of the angle. To exhibit this more explicitly, we substitute 0 
a = id, in (XIII) and obtain 

expresses 
the cosine 
= 2a, or 

2 cos 2 1 = 1 + cos 0; 

n 

2 sin 2 - = 1 — cos 0. 

A 

(XVI) 

Or, cos! « = l + “s«. 

. „ 6 1 — cos 9 

sin2 2 = 2 • 

(1) 


On extracting square roots in (1) we obtain (XIV). 














108 


TRIGONOMETRY 
By use of (XIV) we obtain 


Proof of (XV). 


tan ^ 


Illustration 1. 


. 0 


/l — cos 0 

sm 2 

2 

0 ~ 
cos ^ 


1 1 -j- cos 0 


By use of (XIV), cos 60° = 


V I — cos 0 
1 + cos. 0 


1 1 + cos 120° /1 - i_ 1 

\ 2 “ \ 2 2 ' 


EXERCISE 45 

By use of (IX) to (XVI), find the functions of the first angle by use of func¬ 
tions of the second angle. Any result may be left in a form involving radicals. 


1 . 

60°; by use of 30°. 

9. 

135°; 

by use of 270°. 

2 . 

120°; by use of 60°. 

10 . 

120°; 

by use of 240°. 

3. 

300°; by use of 150°. 

11 . 

— 135°; by use of — 270°. 

4. 

240°; by use of 120°. 

12 . 

- 45 c 

by use of - 90°. 

5. 

— 60°; by use of — 30°. 

13.* 

90°; 

by use of 45°. 

6 . 

— 120°; by use of — 60°. 

14.* 

270° 

; by use of 135°. 

7. 

30°; by use of 60°. 

15.* 

90°; 

by use of 180°. 

8 . 

60°; by use of 120°. 

16.* 

180° 

; by use of 360°. 

If 0 lies between - 90° and 90°, find 

the functions of 0/2, under the given 


conditions: 


17. sin 0 = *. 18. sin 0 = - *. 19. sin 0 = - f. 20. sin 0 = f. 

By use of one of (IX) to (XVI), write a right member without radicals 
which involves only one function of some angle: 


21 . 

2 

sin 35° cos 35° = ? 

27. 

2 

cos 2 15° = ? 

22 . 

cos 2 65° — sin 2 

65° = ? 

28. 

2 

sin 2 20° = ? 

23. 

1 

- 2 sin 2 80° 

_ ? 

29. 

1 

— cos B = ? 

24. 

2 

cos 2 50° - 1 

= ? 

30. 

1 

+ cos B — ? 

25. 

1 

+ cos 40° = 

? 

31. 

cos 2 H — sin 2 H — 

26. 

1 

- cos 50° = 

? 

32. 

1 

+ cos 2B = ? 


33. 

2 tan 40° 9 

35. 

1 - cos 140° _ 9 

37. 

1 — cos 0 _ ^ 

1 - tan 2 40° ' 

1 + cos 0 

34. 

cot 2 50° - 1 9 

2 cot 50° 

36. 

/1 + cos 100° _ 9 

38. 

1 + cos B _ 9 
1 — cos B 


* In (XI), (XII), or (XV), a zero denominator occurs when the function in the left 
member does not exist. 






















ADDITION FORMULAS AND RELATED TOPICS 


109 


Prove each identity: 

cot 0 — 1 _ 1 — sin 26 
cot 0+1 cos 26 


40. cot x — 


sin 2x 
1 — cos 2x 


Solution of Problem 39. By (IX) and (X„), 
_ sin 2 6 + cos 2 6 — 2 sin 6 cos 6 
cos 2 6 — sin 2 6 

_ _ (cos 6 — sin 0) 2 _ 

(cos 6 — sin 6) (cos 6 + sin 6) 

(Divide numerator and denominator by sin 6.) 


1 — sin 2 6 _ 1 — 2 sin 6 cos 6 
cos 2 6 cos 2 6 — sin 2 6 


(Since 1 = sin 2 6 + cos 2 6) 


cos 6 — sin 6 
cos 6 + sin 6 
cos 6 _ i 
sin 6 
cos 6 
sin 6 


cot 0 — 1 
cot 0 + 1 


41. 

tan x = 

sin 2x 

44. 

1 + cos 2x 

1 — cos 2x 

42. 

tan x = 

45. 

sin 2x 

43. 

cot x — 

1 + cos 2x 
sin 2x 

46. 

47. 


1 


1-2 sin 2 x 
2 

1 + cos 2x 


48. sec 2 0 cos 20 = sec 2 0 — 2 tan 2 0. 

49. 2 cos 0 — cos 20 sec 0 = sec 0. 


50. 

CSC 2 X — 

2 


1 — cos 

2x 

51. 

sec 2 x 

1 


4 sin 2 x 

sin 2 2x 


52. 

sec 2x = 

1 


cos 2 x — 

sin 2 x 

53. 

tan 2a 

cot 2 a 


2 tan a 

cot 2 a — 

1 

54. 

tan 2a = 

2 


cot a — 

tan a 


55. 


56. 


57. 


cot 20 = 
sec 2 a = 
sec 2a = 


esc 0 — 2 sin 0 
2 cos 0 
sec 2 a 
1 — tan 2 a 
esc 2 a 
cot 2 a — 1 


1 — tan 0 _ 1 — sin 20 
1 + tan 0 cos 20 
cos 20 _ cot 0 + 1 

1 — sin 20 cot 0 — 1 


■kExpress the functions 
second angle: 

60. a; in terms of 2a. 

Hint for Problem 60. Since a = 

62. 4a; in terms of 2a. 

63. 6a; in terms of 3a. 

64. 2 A; in terms of 4A. 

65. 4x; in terms of 8x. 

66. 8x; in terms of 4x. 


in terms of the functions of the 

61. 2 B; in terms of B. 

(2a), we use (XIV) and (XV) with0 = 2a. 

67. lOx; in terms of 5x. 

68. \A ; in terms of A. 

69. 3 A; in terms of 6A. 

70. a; in terms of fa. 

71. 3a; in terms of fa. 


of the first angle 































110 


TRIGONOMETRY 


^Express the first function in terms of the second function: 

72. sin 3x; in terms of sin x. 73. cos 3x; in terms of cos x. 

Hint for Problem 72. sin 3x = sin (x + 2x) = sin x cos 2x + cos x sin 2x. 

74. sin Ax; in terms of sin x. 76. cot 3x; in terms of cot x. 

75. tan 3x; in terms of tan x. 77. cos Ax; in terms of sin x. 

83. Product formulas. We shall prove that, for all values of a 


and jS, 

2 sin a cos /3 = sin (a + /3) + sin (a — \ 3); (XVII) 

2 cos a cos /3 = cos (a + P) + cos (a — /3); (XVIII) 

2 sin a sin /3 = cos (a — /3) — cos (a + P ); (XIX) 

2 sin cos a = sin (a + /8) — sin (a — /3). (XX) 

Proof. 1. Recall formulas I, II, V, and VI: 

sin (a + /3) = sin a cos j8 + cos a sin (1) 

sin (a — (8) = sin a cos /3 — cos a sin /3; (2) 

cos (a + /3) = cos a cos j3 — sin a sin |8; (3) 

cos (a — /3) = cos a cos j8 + sin a sin j8. (4) 


2. Add corresponding sides of (1) and (2) to obtain (XVII). Subtract 
each side of (3) from the corresponding side of (4) to obtain (XIX); etc. 

Note 1. In words, (XVIII) states that twice the product of the cosines of two 
angles equals the sum of the cosines of the sum and the difference of the angles. Similar 
statements should be given for (XVII) and (XIX). 

Example 1. Check the truth of (XIX) if a = 150° and /3 = 30°. 
Solution. 1. From (XIX), 2 sin 150° sin 30° = cos 120° — cos 180°. 

2. We compute each side: 2 sin 150° sin 30° = 2(^)( 5 ) = 5 . 

3 . cos 120° — cos 180° =—£—(— 1) = h This checks with Step 2 . 

Example 2. Express as a sum: 2 sin 2 6 cos 89. 

Solution. By use of (XVII) with a = 29 and /3 = 8 9, 

2 sin 29 cos 8 9 = sin 100 + sin (— 60 ) = sin 100 — sin 60 . 

Note 2. By use of Section 55, page 68 , 

sin (a - j3) = sin [— (j3 - a)] = — sin (/3 — a). 

Hence, (XX) can be written 

2 sin /3 cos a = sin (a + jS) + sin (/3 — a). (5) 

Notice that (5) is the same as (XVII) except for notation: interchange a and /3 
in (XVII) and (5) is obtained. Hence, (XX) need not he remembered because it 
is equivalent to (XVII). 


ADDITION FORMULAS AND RELATED TOPICS 111 
84. Sums and differences of sines or of cosines. In (XVII) to 


(XX), let 

x = a + 0 and y = a — 0. 

(1) 

Then, 

x + y = 2a and x — y = 20. Hence, 



a. = K* + y)\ 0 = Kx - y ). 

(2) 

On substituting (1) and (2) in (XVII) to (XX), we obtain 



, • n • * + W X — U 

sm x + sin y = 2sin — cos — 2 ’ 

(XXI) 


n x + w . x — y 

sm x - sin y =2 cos —~ sm —; 

(XXII) 


_ x + y x — y 

cos x + cos y = 2 cos — cos — ; 

z z 

(XXIII) 


X -4- v x — V 

cos x — cos y = — 2 sin — sin —• 

(XXIV) 


Note 1. Formulas XXI and XXII are particularly necessary for the develop¬ 
ment of numerical trigonometry. 

Illustration 1. By use of (XXI), 

sin 50° + sin 10° = 2 sin 4(60°) cos £(40°) = 2 sin 30° cos 20°; 

sin 20° + sin 80° = 2 sin 50° cos (— 30°) = 2 sin 50° cos 30°. [p. 68] 

Comment. Whenever a function of the negative of an angle is met in a result, 
use Section 55, page 68, to change to the corresponding positive angle. 

Illustration 2. By use of (XXIII), 

cos 4 A + cos 2 A = 2 cos £(4 A + 2 A) cos 4(4 A — 2 A) == 2 cos 3A cos A. 

_ . . _ ,, ., ... cos 5a + cos 3a , 

Example 1. Prove the identity: -—= - ; — 5 - = tan a. 

^ sin 5a — sin 3a 

Solution. Apply (XXIII) in the numerator and (XXII) in the denominator: 

cos 5a + cos 3a 2 cos 4a cos a cos a _ A1?r , 

—; —■ r ~ A ; — . — tSill Oi. 

sin 5a — sm 3a 2 cos 4a sin a sin a 

EXERCISE 46 

Check formulas XVII, XVIII, and XIX for each pair of angles: 

1. a = 60°; 0 = 30°. 4. a = 90°; 0 = 45°. 7. a = 60°; 0 = 300°. 

2. a = 120°; 0 = 60°. 5. a = 30°; 0 = 60°. 8 . a = 30°; 0 = 180°. 

3. a = 300°; 0 « 60°. 6 . a = 60°; 0 = 120°. 9. a = 45°; 0 = 270°. 

Express as a sum or a difference of functions of positive multiples of 6: 

10. 2 sin 30 cos 5 9. 14. 2 sin 29 cos 5 9. 18. 2 sin 39 cos 9. 

11. 2 sin 30 sin 59. 15. 2 cos 39 sin 29. 19. 2 cos 39 cos 29. 

12. 2 cos 39 cos 50. 16. 2 cos 0 cos 50. 20. 2 cos 50 sin 0. 

13. 2 cos 50 cos 70. 17. 2 sin 0 sin 70. 21. 2 sin 0 sin 90. 














112 


TRIGONOMETRY 


Express 

each 

sum or difference as a product: 





22 . 

sin 

60° 

- sin 20°. 

26. 

cos 

40° 

- 

cos 80°. 

23. 

cos 

80° 

— cos 30°. 

27. 

sin 

30° 

— 

sin 110°. 

24. 

sin 

40° 

+ sin 30°. 

28. 

cos 

40° 

+ 

cos 140°, 

25. 

cos 

150 

0 + cos 130°. 

29. 

sin 

60° 

+ 

sin 140°. 


30. 

sin 3x + sin x. 

34. cos 5x + cos 7x. 38. sin 2 y 

— sin 4 y. 

31. 

cos 4x + cos 2x. 

35. sin 3x + sin 5x. 39. cos 5 y 

— cos 2 y. 

32. 

sin 4x — sin 2x. 

36. cos 2x 

— cos 

4x. 40. sin 7y + sin 9 y. 

33. 

cos 3x — cos x. 

37. sin x — 

sin 3x. 41. cos 3 y + cos 8 y. 

Prove each identity: 





42. 

sin 5x + sin 3x 
sin 5x — sin 3x 

tan 4x 

tan x 

48. 

cos x + cos 9x 
sin x -f sin 9x 

= cot 5x. 

43. 

sin 4x — sin 2x 

tan x 

49. 

sin 3x + sin 7x 

= tan 5x. 

sin 4x + sin 2x 

tan 3x 

cos 7x + cos 3x 

44. 

cos 3x + cos x 

cot 2x 

50. 

sin 3x — sin 5x 

cot 4x. 

cos 3x — cos x 

tan x 

cos 5x — cos 3x 

45. 

cos 4x — cos 2x _ 

tan 3x 

51. 

cos 2x — cos 6x 

= tan 4x. 

cos 4x + cos 2x 

cot X 

sin 6x — sin 2x 

46. 

sin 5x + sin 3x _ 

— cot X. 

52. 

sin 2x — sin x 

X 

. tan ^ 

cos 5x — cos 3x 

cos 2x + cos x 

47. 

sin 4x — sin 2x _ 

tan x. 

53. 

cos 5x + cos 2x 

7x 

= cot ~2" 

cos 4x + cos 2x 

sin 5x + sin 2x 


Express as a product involving only tangents or cotangents: 


54. 

sin 75° + sin 15° 

56. 

sin 6 A + sin 7A 

58. 

sin 75° — sin 15° 

cos 6A + cos 7A 

55. 

cos 75° + cos 15° 

57. 

sin 5 B — sin 4 B 

59. 

cos 75° — cos 15° 

sin 5 B + sin 4 B 


cos a + cos /3 
sin a + sin /3 
sin a + sin /3 
cos a — cos 


85. Miscellaneous identities.* In proving identities, 

A. if possible, avoid introducing radicals; 

B. usually it is best to express any secant or cosecant of a complicated 
angle in terms of its sine or cosine; 

C. if functions of more than one angle appear, it may be convenient to 
express all functions in terms of functions of a single angle. 

Example 1. Prove the identity: cos 3x = cos x — 4 sin 2 x cos x. 
Solution, cos 3x = cos (x + 2x) = cos x cos 2x — sin x sin 2x 
= cos x (1 — 2 sin 2 x) — sin x(2 sin x cos x) 

= cos x — 4 sin 2 x cos x. 

* In a brief course the instructor may desire to omit the remainder of the chapter. 
























ADDITION FORMULAS AND RELATED TOPICS 


113 


Example 2. Prove the identity: -~ ° S * + SIn x = C os 2x 

sec Sx esc 3a; 

Solution. Since sec 3a; = — ^ and esc 3x = — ~ . 

cos 3x sin 3a; 

cos x sin x 

sec3x + esc 3x = cos * cos 3x + sm x sin 3x = cos ~ *) = cos 2s. 


Example 3. Prove the identity: 

sin 30 — sin 0 = 2 cos 2 0 sin 0 — 2 sin 3 0. 

Solution. By use of (XXII) and (X„), sin 30 - sin 0 = 2 cos 20 sin0 

= 2 (cos 2 0 - sin 2 0) sin 0 = 2 cos 2 0 sin 0 - sin 3 0. Q.E.D. 

Example 4. Prove the identity: cot | - 

2 1 — cos 0 

Solution. By use of (IX) with a = id, and (XVI), 


sin 0 


0 . 0 0 0 
2 sin - cos ^ cos ^ 


1 — cos 


2 sin* | 


“2 


= cot 2* Q.E.D. 


Comment. A more complicated solution would have resulted if we had ex¬ 
pressed cot id in terms of functions of 0 by use of (XV). Notice that a function 
of any angle is more conveniently expressible in terms of functions of half the angle 
than in terms of functions of twice the angle. 


EXERCISE 47 

Prove each identity: 

1. tan a sin 2a = 2 sin 2 a. 3. 2 cos a = esc a sin 2a. 

2 . cot a sin 2a = 1 + cos 2a. 4. (sin a — cos a ) 2 = 1 — sin 2a. 

5. cos 30 + cos 0 = 4 cos 3 0 — 2 cos 0. 


6. f (cos 30 — cos 50) = sin 40 sin 0. 8. sin 3a; = 3 cos 2 a; sin a; — sin 3 x. 

7. ^ (sin 70 — sin 50) = cos 60 sin 0. 9. sin 4a; = 4 sin x cos x cos 2a;. 

10. cos ( x + \tt) cos (x — iir ) + sin (x + ix) sin (a; — iir) = %. 


11 . 

12 . 


sm x . cos x . „ 

H-— = sm 3a;. 


sec 2a; 
cos 2a: 


esc 2a; 
sin x 


sec x 

13. cot 3a 

14. sec 2a = 


= cos 3a;. 


esc 2a; 

1 — 3 tan 2 a 
3 tan a — tan 3 a 
_sec a_ 

cos- a — sin a tan a 


15. esc 2a 

^ 2 tan a 

tan 2a 


tan a 
2 sin 2 a 

1 — tan 2 a. 


17. sec 2a 

18. 2 tan a 


sec a _ 

2 cos a — sec a . 
1 — tan 2 a 


19. tan 2 0 + cos 20 = 1 — cos 20 tan 2 0. 


cot 2a 






















114 


TRIGONOMETRY 


21 . cot 2 a - 

22 


_ _ cos 3x sin x oo 5 0 />» 

20 .- jr— = cos 2 2a; — sin 2 lx. 

sec x esc ox 

cos a — sin a tan a nA 1 + cot x 1 + sin 2a; 


2 sin a 
esc x — sec x cos 2a; 


sec x + esc x 1 + sin 2a; 

cos 2a; — cos x 


23 . cos x — 1 = 


2 cos x + 1 


24 . 

25 . 

26 . 


cot x — 1 cos 2a; 
sin x + sin 3a; 2 cot x 


cos x + cos 3a: cot 2 a; — 1 
cos x cos 3a: 


esc 3a; esc x 


= sin 2a:. 


27 . cos ( a + (3) cos (a - 0) = cos 2 0 - sin 2 a. 

28 . sin 3a: cot x + cos 3x = sin 4a; esc x. 

29 . cos 0 — sin 0 tan 26 = cos 30 sec 20. 

tan 2x 2 tan x 


30 . 

31 . 

32 . 


1 + tan 2a; 

cot 2a; 

1 — cot 2x 

sec (a + 0) = 


1 + 2 tan x — tan 2 x 

cot 2 x — 1 

■ ■ — • 

2 cot x + 1 — cot 2 x 

sec a sec 0 
1 — tan a tan 0 
esc a esc 0 


33 . esc (a — 0) = , o , 

cot 0 — cot a 

34 . sec 3a; sin 6a: = 2 tan 3a; cos 3a:. 37 . 2 cos 2a = esc 2a sin 4 a. 

35 . 2 cot 4a; = cot 2x — tan 2a;. 38 . sin 6a tan 3a = 2 sin 2 3a. 

36 . cos 4a; sec 2 2a; = 1 — tan 2 2a;. 39 . tan 6a sec 3a = 2 sin 3a sec 6a. 

40 . sec 2a cos 4a = cos 2a — sin 2a tan 2a. 

41 . 1 + cos 4a; = 2 cos 2x cos 2 x — 2 cos 2a; sin 2 x. 


42 . sec 2 2x — 1 = 

44 . 


1 — cos 4a; 


43 . 


1 + cos 6x 


esc 2 3a: — 1. 


1 + cos 4a; 1 — cos 6a; 

tan (a — 0) + tan 0 _ cot 0 — cot (a + 0) 

1 — tan (a — 0) tan 0 cot (a + 0) cot 0 + 1 

45 . sin (a —0) cos 0 + cos (a —0) sin 0 = sin (a+0) cos 0 —cos (a + 0) sin 0. 

.. sin a; . x 

48 . -= tan — 

1 + cos x 2 


.. X _ X „X A 

46 . sm x esc s = 2 cos s • 47 . esc 2 = - - 

2 2 2 1 — cos x 


49 


COS X 


cos hx + sin kx 


= cos \x — sin hx. 


n XX 

50 . 2 esc x = sec ^ esc ^ 


51 


cos 3 x + sin 3 x 
2 — sin 2a; 


• £ £ 
52 . 2 cot x = cot 0 — tan -• 


— \ sin x + \ cos x. 

1 + COS X 


X 


X 


X 


53 . cot 2 ^ — 1 = esc 2 ^ cos x. 


54 . " 1 "" - = esc 2 7,-1. 
1 — cos x 2 

__ 1 + sin x tan %x + 1 

Wwi 1 i 1 

cos x 1 — tan \x 


55 . 


COS X 































ADDITION FORMULAS AND RELATED TOPICS 


115 


SUPPLEMENTARY PROBLEMS 

Prove each identity: 

cos x cot \x — 1 


56.-. - 

1 + sm x cot \x + 1 

58. 2 sin 3a; cos 2a; — sin a — sin 5a:. 


57. sec 2x = 


cos 4 x 


sm 4 x 


59. cos 4a = cos 2a — 2 sin 3a: sin a. 
2 cos 2a: 


60. 1 + cot a cot 3a: = --—- 

cos 2a: — cos 4 a 

61. 4 cos 6a; sin 2a: cos 4x = sin 4x — sin 8a: + sin 12a;. 


Solution, (sin 4x — sin 8x) + sin 12a: 

= 2 cos 6x sin (— 2a;) + sin 12a: [By use of (XXII)] 

= — 2 cos 6x sin 2a: + 2 sin 6a; cos 6a: [By use of (IX)] 

= 2 cos 6x (sin 6a; — sin 2x) = 4 cos 6a; cos 4a: sin 2a;. 


62. 

sin 

10a; 

— sin 6x + sin 4x 

= 4 sin 2x cos 3x cos 5x. 


63. 

4 cos 3x 

sin 2x sin 5x — 1 

= cos 6.r — 

cos 4x — cos lOx. 


64. 

sin 

5a; - 

- sin 3x + sin 2x = 

= 4 sin x cos 

fx cos fx. 


65. 

cos 

4a: = 

= 8 cos 4 a + 8 sin 2 

a — 7. 



66. 

sin 

5a: = 

= 16 sin 5 a — 15 sin a + 20 sin 

a cos 2 a. 


67. 

sin 

4a: + 4 sin 2a: sin 2 a = 

2 sin 2a:. 



68. 

cos 

x + 

cos (x + y) + cos 

(x + 2 y) = 

cos (x + y) sin f y esc 

\y- 

69. 

sin 

x — 

sin (x + y) + sin 

(x + 2 y) = 

sin (x + y) cos f y sec 

\y■ 


Prove each identity without altering the right member: 

70. 4 sin 3 0 = 3 sin 9 — sin 30. 71. 4 cos 3 0 = 3 cos 0 + cos 30. 

Hint for Problem 70. 4 sin 3 0 = 2 sin 0(2 sin 2 0). Then use (XIII). 

72. 16 cos 5 0 = 5 cos 30 + 10 cos 0 + cos 50. 

73. 64 sin 7 x = 35 sin x — 21 sin 3a; + 7 sin 5x — sin 7x. 


86. Miscellaneous equations. An equation in x which explicitly 
states the value of one function of some multiple of x should be 
solved without alteration, by inspection or perhaps with the aid of 
a trigonometric table. 

Example 1. Solve; sin 3a; = \. 

Solution. 1. Recall that sin 30° = i = sin 150°. Either of these angles, plus 
any integral multiple of 360°, is a value of “3a;” which satisfies the equation. 
Hence, we obtain 3a; = 30°, 390°, 750°; 150°, 510°, 870°, etc. 

2. If 3a; = 30°, then x = 10°; if 3x = 390°, then x = 130°; etc. The solu¬ 
tions between 0° and 360° are x = 10°, 50°, 130°, 170°, 250°, 290°. 

Comment. We needed all values of 3a: between 0° and 3(360 ) so that, after 
dividing by 3, we would have all values of x between 0° and 360°. 






116 


TRIGONOMETRY 


To solve an equation, we aim to find one or more equations, each 

involving only one function of one angle , whose solutions include all 
solutions of the given equation. 

Example 2. Solve: cos 2x — cos x = 0. 

Solution. 1. Use (X*) so that only functions of x will remain: 

2 cos 2 x — 1 — cos x = 0; or, (2 cos x + l)(cos x — 1) = 0. 

2. Solve 2 cos x + 1 = 0 and cos x — 1 =0: x = 120°, 240°, and 0°. 

Example 3. Solve: sin | 

Solution. 1. Recall that sin 60° = ^^3 = sin 120°. 

2. Hence, = 60° or 120°. Therefore, 0 = 120° or 240°. 

Example 4. Solve: sin 5x — sin x = cos 3a;. 

Solution. 1. Use (XXII): 2 cos 3a; sin 2a; = cos 3a;. 

2 cos 3a; sin 2x — cos 3a: = 0; cos 3x (2 sin 2a; — 1) = 0. 

2. Hence, cos 3a; = 0 or 2 sin 2a; — 1 = 0. The student should complete 
the solution of each of these equations as in Example 1. 

Note 1. Suggestions A and B of page 97, for simple equations, and remarks 
A, B, and C of page 112 should be kept in mind when solving an equation. 








EXERCISE 48 





Solve 

each equation :* 









1. 

pin 

2x 

= 0. 


4. 

cos 

2a; = 

*V3. 

7. 

sin 3a; = 

^V3- 

2. 

sin 

2x 

_ 1 

— 2 • 


5. 

sin 

3a; = 

- 1. 

8. 

cos 3a: = 

- 

\V2. 

3. 

cos 

2x 

= - 1. 


6. 

CSC 

2a; = 

- 1. 

9. 

cot 4a; = 

- 

V3. 

10. 

tan 

4a; 

= Vs. 

11. 

cot; 

3x - 

= 1. 

12. cos 

3a; = 

0. 13. 

sec 

2a; = 

14. 

sin 

X 

2 " 

_ i 
" 2* 


16. 

tan 

X 

2 ~ 

- 1. 

18. 

tan | = 

V3 


15. 

cos 

X 

2 ~ 

1 

2 • 


17. 

cot 

X 

2 ~ 

1. 

19. 

sin | = 

V3 

2 


20. 

sin 

2x 

= 2 sin 

X. 

22. 

cos 

2a; = 

sin x. 

24. 

sin 2a; = 

: — 

sin x. 

21. 

cos 

x = 

= sin 2x. 


23. 

cos 

2a; = 

— cos a:. 

25. 

cot 2a; = 

= - 

cot X. 

26. 

sin 

x + cos 2a: 

- 1 

= 0. 



29. 2 - 

3 sin x — cos 2a:. 


27. 

cos 

2a; 

— COS X 

+ 1 

= 0. 



30. cos 2a; + 2 

cos 2 x = 

2. 


28. 

cos 

2.r 

— 2 sin 2 

x — 

2. 



31. cos 2a; — 3 

cos x + 

2 = 

■■ 0. 


* We desire all positive or zero solutions less than 360°. If none exist less than 
360°, find one positive solution if there are any solutions. 


ADDITION FORMULAS AND RELATED TOPICS 


117 


32. 

cos 2a; 

= COS 2 X. 

37. 

sin 

4a: = 

= 3 cos 2a;. 

42. 

4 cos 2 3a: 

= 

3. 


CO 

CO 

cos 2a; 

= — sin 2 x. 

38. 

cos 

4a: = 

= cos 2 

2x. 

43. 

2 — sec 2 

2x 

= 

0. 

34. 

sin 2a; 

= 2 cos 2 x. 

39. 

cos 

4a; = 

= 2 sin 

2 2x. 

44. 

2 — esc 2 

3x 

= 

0. 

35. 

sin 2x 

= 2 sin 2 x. 

40. 

cos 

4a; = 

= cos 

2x. 

45. 

tan 4a; = 

tan 2x. 

36. 

sin 4a; 

= 5 sin 2a;. 

41. 

sin 2 

3a; ■ 

= 3. 


46. 

cot 4a; + 

cot 2a; = 0. 

47. 

2 sin 2 

3a; — 3 sin 3a; 

= 

2. 


49. 

sin 2 : 

2x + 1 

] sin 2x -f 

■ 2 

= 

0. 

48. 

2 cos 2 

2a; + 5 cos 2x 

= 

3. 


50. 

sin 3a; + sin 3a; tan 

2a; 

= 

0. 

51. 

sin ! 

= sin x. 

52. 

sin 

x = 

2 sin 

X 

2 

53. 

COS X = 

sin 

X 

2 



Hint for Problem 51. Use (IX) in the right member: sin x = 2 sin ^ cos 

Z Z 

We prefer not to use (XIV) on the left because it would introduce a radical. 


54. cos x + cos 2=0- 


55. cos x + 2 cos 2 - = 2. 

Z 


56. sin 3a; + sin x = 0. 57. cos 3a: — cos x = 0. 

Hint for Problem 56. Use (XXI). 

58. cos 2x — cos 6a; = 0. 59. cos 2x + cos 3a; = 0. 60. sin 5x = sin x. 
cos 2x — cos 4x = 2 sin 3x. 64. cos 5x + cos x — 2 cos 2x = 0. 


61 


62. sin 2x + sin Ax = sin 3a:. 

63. sin 3x — sin 5x. 


65. cos x — cos 5x + 2 sin 3x = 0. 

66. sin x + sin 5x + sin 3x = 0. 


67. sin x cos 2x + cos x sin 2x = 2. 

68. cos 2x cos x + sin 2x sin x = 1 . 

69. esc | + cot | = 2 sin 70. cot 2 1 + 3 esc | + 3 = 0. 

71. cos 3a: tan 2 3a: = 3 cos 3a;. 72. tan 2 2x + sec 2 2a; = 9. 


SUPPLEMENTARY PROBLEMS 


73. 

5 cos 2a: = 3. 

74. 

5 tan 3x 

= 7. 

75. tan 2 2a; + 

6 

= 5 

76. 

6 sin 2 2a; + 5 

sin 2a; = 

6. 

81. 

cot 3a; + cot x = 

0. 


77. 

78. 

6 sin 2 2a; — 9 

tan 3x cos x - 

sin 2x + 

- sin x = 

3 = 0. 

= 0. 

82. 

tan x + tan 2a; 

1 — tan x tan 2a; 

= 

1. 

79. 

cot 2a; cos x - 

- sin x = 

= 0. 

83. 

cot 3a: cot x + 1 

_ 

< 
CO 1 

80. 

tan 5a: — tan 

3x = 0. 


cot 3a; — cot x 




84. sin 3x + sin 5a; + cos 2x — cos 6a; = 0. 

85. cos 3a: + cos 5x + sin 2a: - sin 6a; = 0. 

87. tan 4a; + 2 sin 2a; = 0. 


86. tan 2a; — 2 sin x = 0. 




118 


TRIGONOMETRY 


REVIEW EXERCISE 49 

Find the functions of the given angle by the specified method, without using 
a table. Rationalize denominators in the results. 

1. 75°; by use of functions of the sum of two angles. 

2. 105°; by use of functions of the difference of two angles. 

3. 15°; use half-angle formulas. 5. 60°; use double-angle formulas. 

4. 22\°\ use half-angle formulas. 6. 120°; use double-angle formulas. 

If a and (3 are positive and acute, and if sin a = f and cos 0 = tt, find 
the functions of the specified angle: 

7. (« + 0). 8. (a — 0). 9. 2a. 10. 2/3. 11. ia. 12. i0. 

Find the functions of 0 and of 0/2 under the given condition: 

13. 0 is acute and sin 0 = i. 14. 90° < 0 < 180°, and sin 0 = *. 


Express each product as a sum and each sum as a product: 


15. 

cos 4a + cos 5a. 

19. 

2 cos 

6° sin 4°. 

23. 

cos 

0 cos 50. 

16. 

sin 5a — sin 3a. 

20. 

2 sin 

10° cos 8°. 

24. 

sin 

3a cos 20. 

17. 

cos 75° — cos 25°. 

21. 

2 cos 

9 sin 

90. 

25. 

sin 

3a + sin 20. 

18. 

sin 35° + sin 55°. 

22. 

2 sin 

29 sin 4 9. 

26. 

cos 

3a — cos 50. 

Prove each identity: 








27. 

2 tan a cot 2a = 1 — 

tan 2 

a. 

29. 

cos 2a esc 

a = 

CSC 

a — 2 sin a. 

28. 

cos 2a sec a = 2 cos a 

— sec a. 

30. 

cos 4 2x — 

sin 4 

2x 

= cos Ax. 


33. 

34. 

35. 

36. 

37. 

38. 


31. sin 2a + cos 2 a tan 3a = sin 5a sec 3a. 

32. 2 cos 0 cot 0 - esc 0 = 2 cos 0 cot 20. 


cos 2x + sin 2x cot 2x -f- 1 


cos 2x — sin 2x cot 2x — 1 

. _ sec 3x 

sin x — ^ _|_ tan Sx 

sin Ax — sin 2x tan x 


-7 39. 


sin 4x + sin 2x 
cos x — cos 3x 


tan 3x 
tan x 


cos 3x + cos x cot 2x 

tan x\ 2 1 + sin 2x 

sin 2x 

esc a 


/ I + tan x \ 2 _ 1 + sii 
\1 — tan x) 1 — sii 


tan x 
sec 2a = 


esc a — 2 sin a 


40. 


41. 


42. 


43. 


ten ^ = 

. e 

cot 2 = 


1 — cos 9 
sin 9 

1 -b cos 9 


sin 9 
sin x — cos 2x 


= 2 sin x — 1. 


sin x + 1 
1 — tan 9 1 — sin 29 


1 + tan 9 
1 + sin 2x 


cos 29 

cos 2x 


cos x + sin x cos x — sin x 
sec |a 


44. sec a = 


2 cos \a — sec §a 


cot (a — 0) cot jS — 1 _ 1 + tan (a + 0) tan 0 

cot (a — 0) + cot 0 tan (a + 0) — tan 0 


45. 






















ADDITION FORMULAS AND RELATED TOPICS 


119 


Prove each identity: 

cos x 


46. + 

sec 2x 


esc 2x 

48. 


CSC 3a; 
sin 2x 
cot x 


47. sin 3a: + 


cos 3a: sin 5a: 


cot 2x 
sin 2x — cos 2x 


cos 2x 


1 — cot x 


Solve each equation: 

49. tan 2x = — V 3 . 

50. sin 2x = \^/ 3. 

51. cos 3a; = — 1. 

52. tan 3a: — V 3 . 

53. cot 2a; = °o. 


54. 

55. 

56. 


sec 

a; 

V2. 

57. 

sec 

2a; 

= 

00 . 


2 “ 

58. 

tan 

co 

= 

00 . 



a; 

< 
to 1 

59. 

sin 

2x 

— 

sin 

X. 

sin 

— = 

-- 








2 

2 

60. 

cos 

2x 

+ 

cos 2 

X 

CSC 

00 

H 

II 

= - 1. 

61. 

cos 

2x 

= 

sin 2 

X. 


62. 

tan 2 2a; + sec 2 2x 

= 9. 

67. 

cos 

4a: + sin 2 2a: = 0. 

63. 

sec 3a: + tan 3a: = 

= 1. 

68. 

cos : 

! 4x + 2 cos 2 2a: = 2. 

64. 

cos 3a: = 1 — sin 

3x. 

69. 

cot 

x — cot 3a; = esc 3a;. 

65. 

sin 2a: — sin 6a: = 

= 0 . 

70. 

cos 

5a; — cos x — 2 sin 2a;. 

66. 

cos x — cos 3a; - 

0 . 

71. 

sin 

3a: + sin 2a; + sin 4a: = 0. 


72. 

i sec 

x sec 2a: = 

tan 

x + tan 2a;. 


73. 

CSC X 

esc 3a: = cot x 

cot 3a: + 1. 

74. 

tan x + tan 2a: 

= 1. 

75. 

cot 

2a; cot 3a: + 1 A /~ 

1 — tan x tan 2x 


cot 2x — cot 3a; 


★ Express each function in terms of the specified functions: 

76. sin (a + /3 — 7 ); in terms of the sines and cosines of a, (3, and 7 . 

77. tan (a — /3 — 7 ); in terms of the tangents of a, (3, and 7 . 

78. cos 4a;; in terms of cos x. 79. sin 3/3; in terms of sin /3. 


★ // a + /3 + 7 = 180°, prove the identity: 

80. sin a + sin /3 — sin 7 = 4 sin \a sin i/3 cos £ 7 . 

Hint. 1. By use of formulas IX and XXI, 

(sin a + sin /3) - sin 7 = 2 sin i(a + 0) cos i(a - 13) - 2 sin ^7 cos § 7 . 

2. Since 7 = 180° - (a + (3) and £7 = 90° - \{a + 13), hence 
(A) sin §7 = cos Kct + /3) and (R) sin §(« + /3) = cos I 7 . 

3. Substitute the right members of (A) and (R) in Step 1 and use (XXIV). 

81. sin a + sin /3 + sin 7 = 4 cos \ol cos i/3 cos § 7 . 

82. cos a + cos j 8 + cos 7 = 1 + 4 sin i<x sin i/3 sin £ 7 . 

83. cos a + cos /3 — cos 7 = 4 cos ia cos i/3 sin 57 — 1. 

84. tan a + tan /3 + tan 7 = tan a tan (3 tan 7 . 

85. sin 2 a + sin 2 /3 + sin 2 7 = 2 + 2 cos a cos /3 cos 7 . 











CHAPTER IX 


OBLIQUE TRIANGLES 


87. Introduction. In any triangle ABC, let a, (3, and y be the 
angles at A, B, and C, and let a, b, and c be the lengths of the cor¬ 
responding opposite sides. The angles (a, j 6 , y) and the sides 
(a, b, c ) will be called the six parts of the triangle. 

By use of plane geometry, we can con- c 

struct a triangle if we are given any three & 
of its parts, of which at least one is a side. 7 

Data of this nature fall into four different a c n 

classes. Fig. 73 


Case I. 
Case II. 
Case III. 
Case IV. 


Given two angles and a side. 

Given two sides and an angle opposite one of them. 
Given two sides and the included angle. 

Given three sides. 


In this chapter we shall develop methods for computing the un¬ 
known parts of a triangle under each of Cases I to IV. The computa¬ 
tion of the unknown parts is called the solution* of the triangle. 


88. The law of cosines. In any triangle, the square of any side is 
equal to the sum of the squares of the other sides minus twice their 
product times the cosine of their included angle. That is, 

a 2 = & _j_ C 2 _ 2 be cos a; (1) 

b 2 = a 2 _ C 2 _ 2 ac cos /3; (2) 

c 2 = a 2 + b 2 — 2ab cos y. (3) 


On solving (1), (2), and (3) for the cosines, we obtain 


b 2 + c 2 - a 2 


cos /? = 


a 2 + c 2 - b 2 


cos y = 


a 2 + b 2 - c 2 


(4) 


2 be ’ — K 2ac ’ ' 2 ab 

Note 1. Suppose that triangle ABC is a right triangle with y — 90°. Then, 

cos y = l, cos a = -» and cos B = -• Hence, (1), (2), and (3) become 
c c 

a 2 = fe 2 + C 2 _ 2 b\ or o 2 = c 2 - b 2 ; b 2 = c 2 - a 2 ; c 2 = a 2 + b 2 . 

Thus, for a right triangle, the law of cosines is equivalent to the Pythagorean theorem. 


* We shall find that, if the data violate certain conditions, then no solution will 
exist. Under Case II we shall sometimes find two solutions. Otherwise, there will 
always be exactly one solution. 


120 






OBLIQUE TRIANGLES 


121 


Proof of the law of cosines. 1. Let a be any side and let /3 be an 
acute * angle of the triangle. Drop a perpendicular CD from C to AB, or 




AB extended. One obtains Figure 74 if a is acute and Figure 75 if a is 


obtuse. In either figure, let 

AD = m and DC = h. 

2. From triangle ADC, b 2 = h 2 + m 2 ; h 2 = b 2 — m 2 . (5) 

3. From triangle DBC, a 2 = h 2 + (DB ) 2 . ( 6 ) 

Substitute (5) in ( 6 ): a 2 = b 2 — m 2 + (DB) 2 . (7) 

4. Suppose that a is not obtuse; then, in Figure 74, 

DB = AB — AD; hence, DB = c — m. ( 8 ) 

171 

From triangle ADC, cos a = m = b cos a. (9) 

Substitute ( 8 ) in (7): a 2 = b 2 — m 2 + (c — m) 2 ; 

o 2 = b 2 +-c 2 — 2cm = 6 2 + e 2 — 2 be cos a. [Using (9)3 

5. Suppose that a is obtuse; then, in Figure 75, 

~DB = AB + AD; hence, DB = c + m. (10) 

7Y1 

From triangle ADC, cos 6 = > and a = 180° — 9. From page 70, 

cos a = — cos 9, or cos a = — m — — b cos a. ( 11 ) 

Substitute (10) in (7): a 2 = 6 2 — m 2 + (c + m) 2 ; 


(i 2 — b 2 c 2 ”f~ 2 cm = b 2 + c 2 — 2 be cos a. [Using (11)] 

6. Hence, we have proved (1) for any triangle. Since a represented 
any side in our proof, we may immediately write down ( 2 ) by changing a 
to b, and ( 3 ) by changing a to c, with corresponding changes in the other 
letters. 


At least one angle other than a is acute because a + /3 + 7 = 180°. 






122 


TRIGONOMETRY 


★Note 2. In Figure 74, c = DR + m, or 

c = a cos /3 + & cos a. [Using (9)3 

In Figure 75, c = DR — m = a cos jS + 6 cos a. [Using (11)] 

It follows that, for any triangle, 

c = a cos j3 + b cos a; a = b cos y + c cos jS; b = a cos 7 + c cos a. (12) 
Formulas 12 are sometimes useful in checking the solution of a triangle. 

89. The use of the law of cosines in solving triangles is restricted 
because its formulas are not suited for logarithmic computation; 
each of the formulas involves a sum of terms, and logarithms do not 
simplify the computation of a sum. However, if a computing ma¬ 
chine is available or if convenient numbers are involved, the law of 
cosines is very useful in the solution of triangles without logarithms. 

Solution of Case IV by use of the law of cosines: 

1. Find the angles by use of formulas 4, Section 88. 

2. Check the solution by use of a + /3 -j- 7 = 180°. 

Example 1. Solve triangle ABC if a = 5, b = 7, and c — 11. 

Solution. 1. By use of formulas 4, Section 88, 

cos a = i-M = .9416; cos /3 = ^ = .8818; cos 7 = - U = ~ -6714. 

2. By use of Table VII, a = 19° 4U; /3 = 28° 8'. 

3. Since cos 7 is negative, 7 is obtuse. Let 7 = 180° — 6; then Q is acute 
and cos Q = .6714. From Table VII, 6 = 47° 50'. Hence, 

7 = 180° - 47° 50' = 132° 10'. (Recall page 65) 

Check, ot + p + y = 19° 41' + 28° 8' + 132° 10' = 179° 59'; satisfactory. 

It is possible to solve a triangle under Case III entirely by use of 
the law of cosines but other formulas will prove more convenient. 
Hence, we shall restrict ourselves here to finding only the third side if 
two sides and the included angle of a triangle are given. 

Illustration 1. If a = 5, b = 12, and 7 = 60°, then from (3), page 120, 

c 2 = 25 + 144 - 2(5)(12) cos 60° = 109. (cos 60° = *) 

By use of logarithms, or Table I, c = Vl09 = 10.4. 

EXERCISE 50 

Suppose that a is obtuse. Find a by use of Table VII: 

1. sin a = .2447. 3. cos a = — .1363. 

2. sin a = .9636. 4. cos a = - .7969. 

Note. Solve each of the remaining problems by use of the law of cosines; use 
Table VII unless otherwise directed. 


OBLIQUE TRIANGLES 


123 


Solve, without trigonometric tables if possible: 


5. a = 3; b = 2; y = 60°: find c. 

6 . b = 4; c = V^a: = 30°: find a. 

7. b — V2;a = 8 ; 7 = 45°: findc. 

8 . a = V2;c = 5;0 = 45°: find 6 . 

9 . b = 7; c = V2; a = 135°: finda. 

10 . a = 2 ; 6 = 2 ; 7 = 120 °: find c. 

11. c = 9; 6 = V3; a = 150°: find o. 

12. a = V3;c =4; /3 =150°: find 6 . 


13. a = 3; 6 = 10; c = 8 : find y. 

14. a = 7; 6 = 9; c = 4: find a. 

15. a = 5; 6 = 6 ; c = 7: find /3. 

16. a = 9; 6 = 10; c = 7: find 7 . 

17. a = 13; 6 = 7; c = 8 : find a. 

18. a = 6 ; 6 = 12 ; c = 9 : find/ 3 . 

19. a = 12; 6 = 5; c = 15: find 7 . 

20. a = 5; 6 = 15; c = 18: find 7 . 


<Sofoe triangle ABC and check: 

21. a = 5; 6 = 6 ; c = 4. 

22. a = 8 ; 6 = 5; c = 7. 

23. a = 6 ; 6 = 14; c = 10. 


24. a = 13; 6 = 6 ; c = 9. 

25. a = 10; 6 = 11; c = 17, 

26. a = 30; 6 = 15; c = 20, 


Find the specified side; use logarithms for computing any square root. 

27. a = 5; 6 = 7; 7 = 32°: find c. 30. a = 6 ; c = 5; /3 = 49° 17': find 6 . 

28. 6 = 6 ; c = 9; a = 53°: finda. 31. 6 = 4; c = 11; a = 63°28': find a. 

29. a = 5; c = 8 ; jS = 97°: find 6 . 32. a = 9; 6 = 4; 7 = 93° 15': find c. 

33. Prove formula 2, page 120, by use of a figure like Figure 74. 

34. Prove formula 3, page 120, by use of a figure like Figure 75. 

35. Find the sides of a parallelogram if the lengths of its diagonals 
are 12 inches and 16 inches and if one angle formed by the diagonals is 37 °. 

36. Find the length of the diagonal of a parallelogram if one angle 
is 87° and the sides are 10 inches and 15 inches long. 

37. At 1 p.m. a train goes due east from a town T at a speed of 30 miles 
per hour and a second train goes from T in a direction 20° west of south 
at 40 miles per hour. How far apart are the trains at the end of 2 hours? 

38. An enemy's camp at E cannot be seen from a battery of artillery 
at B. An observer at P finds that PE = 7300 yards, PB = 6300 yards, 
and Z.BPE = 41° 27'. Find the distance from the battery to E. 


+Find the magnitude of the resultant of the given forces acting simultaneously 
on the same object: 

39. 200 pounds and 300 pounds: angle between their directions is 78°. 

40. 150 pounds and 250 pounds: angle between their directions is 67°. 

41. 40 pounds and 70 pounds: angle between their directions is 123°. 

42. 25 pounds and 75 pounds: angle between their directions is 146°. 


TRIGONOMETRY 


124 

90. The law of sines. In any triangle , any two sides are propor¬ 
tional to the sines of the opposite angles, that is, 

a _ b b c c _ a ^ 

sin a ~~ sin 0’ sin 0 sin y’ sin y sin a: 


We may abbreviate (1) by writing 
a = b 
sin a sin 0 


A 



c 

sin y 

A 


( 2 ) 



Proof. 1. Let b and c represent any two sides. 

2. Drop a perpendicular AD from A to BC, or BC extended. 

3. Suppose that neither 0 nor y is obtuse, as in Figure 76. 


From triangle ABD, sin 0 = 

h = c sin 0 . 

(3) 

From triangle A CD, sin 7 = 

h = b sin 7 . 

(4) 

From (3) and (4), 

c sin 0 = b sin 7 . 

(5) 

Divide both sides by sin 0 sin 7 : 

c b 

sin 7 sin 0 

(6) 

4. If one of 0 and 7 is obtuse, let 0 be the obtuse angle, as in 

Figure 77. 

Then, from triangle ABD, sin 6 = -• 

Since 0 = 180° — 6, 


sin 0 = sin (180° 

— 6) = sin 6. 


Hence, sin 0 = or 

h = c sin 0 . 

(7) 


We see that (3) and (7) are identical and then we verify that (4), (5), and 
( 6 ) hold true without alteration for Figure 77. Hence, ( 6 ) is true for 
any triangle. 

Note 1. Equations 1 may be rewritten as follows: 

a sin a b _ sin 8 m a _ sin a ^ 

b = sin 0 ’ c — sin y’ c sin 7 ’ 

Also, we may obtain a new form for each equation in (1) or ( 8 ) by inverting the 
fractions in each member. 

















OBLIQUE TRIANGLES 125 

91. Solution of Case I by use of the law of sines. Let the given 
parts of triangle ABC he a , (3, and c. 


1. Find the third angle by use of a + (3 + y = 180°. 

2. Find the unknown sides by use of the law of sines: 

n = c sin a . b - C sin i 8 

sin y ’ sin y ' 


( 1 ) 


3. Optional;* check formula. Use any one of formulas 12, page 122 
which involves y ; for instance, a = c cos 0 + b cos y. This formula is most 
efficient as a check when computed without logarithms. 

★Note 1. The third formula of the law of sines, a/sin a = 6 /sin 0, not used 
in ( 1 ), has moderate value as a check on the solution if both sides are computed 
without logarithms, but is practically worthless if logarithms are employed. In 
the latter case, the formula would not detect an error in any logarithm met in 
computing ( 1 ) because the same logarithms would appear in checking and be¬ 
cause a/sin a = 6 /sin (3 if a and 6 satisfy ( 1 ), even though incorrect values of c, 
a, (3, and y might enter. 


Example 1. Solve triangle ABC if 0 = 37° 6 ', y = 42° 38', and c = 21 . 37 . 


Formulas 

Computation 


Data: c = 21.37; 0 = 37° 6'; y = 42° 38'. 

a = 180° - (0 + y). 

a = 180° - (37° 6' + 42° 38') = 100° 16'. 

6 c 

sin 0 sin y 

or 

ft _ c sin (8 
sin y 

log c = 1.3298 (Table V) 

log sin 0 = 9.7805 - 10 (+) (Table VT\ 

log c sin 0 = 11.1103 — 10 
log sin y= 9.8308 - 10 (-) 

log b = 1.2795; b = 19.03. 

a c 

sin a sin y 

or 

c sin a 

a = — . - 

sm y 

Note, sin 100° 16' = sin (180° - 100° 16') %= sin 79° 44'. 
log c - 1.3298 
log sin a = 9.9930 — 10 (+ ) 
log c sin a = 11.3228 — 10 
log sin y = 9.8308 — 10 ( — ) 

log a m 1.4920; a = 31.04. 


Summary. a = 31.04; b = 19.03; a = 100° 16'. 


Optional: check. c — a cos /3 + 6 cos a. 

cos a = cos 100° 16' = — cos (180° — 100° 16') = — cos 79° 44'. 

a cos 0 = 31.04(.7976) - 24.76. [Using Table VII] 

b cos a = - 19.03 cos 79° 44' = - 19.03(.1782) = - 3.39. 
a cos 0 -f b cos a = 24.76 — 3^39 = 21.37 — > « —c = 21.37; satisfactory. 


* If the instructor desires, checking may be omitted until later formulas are derived. 



























TRIGONOMETRY 


126 


ifComment. Cologarithms may be conveniently used in computing a and b in 
the preceeding Example 1: 


c sin 0 


log sin 7 = 9.8308 - 10: hence, 


c sin a, 
sin 7 ' 


: hence log b — log c + log sin 0 + colog sin 7 . 

log c = 1.3298 
log sin 0 = 9.7805 - 10 

| ( +) 

hence, colog sin 7 = 0.1692 

J 

log b = 1.2795; 

b = 19.03. 

log c = 1.3298 
log sin a = 9.9930 — 10 

}< + ) 

colog sin 7 = 0.1692 

J 

log a = 1.4920; 

a = 31.04. 


EXERCISE 51* 

Solve each triangle without t logarithms by use of Table VII: 

U-5;a - 75°; 0 = 60°. 6. a = 15; 0 = 72°4'; T - 29° 36'. 

2 . c = 3; 0 = 37°; 7 = 60°. 7. b = 12; a = 34°; 0 = 42°. 

3 . c = 25; 0 — 42° 20'; 7 = 64° 30'. 8. c = 100; 0 = 53°; 7 = 19°. 

4. a = 50; a = 37° 30'; 0 = 71° 10'. 9. a = 200; a = 32° 21'; 7 = 21° 39'. 

5 . b = 6; 0 = 53° 25'; a = 48° 15'. 10. b = 35; 0 = 17° 41'; 7 = 52° 19'. 

Solve by use of four-place or five-place logarithms: 

11. c = 15.67; a = 42° 20'; 7 = 53° 40'. 

12. b = 231.6; a = 19° 10'; 0 = 82° 40'. 

13. a = 1.056; 0 = 23° 20'; 7 = 53° 50'. 

14. c = 6019; a = 16° 30'; 0 = 59° 20'. 

15. a = 19° 41'; 0 = 28° 8'; a = 5.37. 

16. a = 64° 9'; 0 = 13° O'; a = 12.3. 

17. a = 23° 54'; 7 = 85° 16'; b = .4317. 

18. 0 = 101° 36'; 7 = 21° 44'; c = .04198. 

19. a = 23° 42'; 0 = 98° 18'; a - .03152. 

20. a = 31° 18'; 7 = 42° 32'; b = .01571. 

21. A polygon is inscribed in a circle whose radius is 6 inches. One of the 
sides subtends an angle of 27° at the center of the circle; find the length 
of this side by use of the law of sines. 

22. One side of a parallelogram is 56 inches long. The diagonals of the 
parallelogram make the angles 35° and 47°, respectively, with this side. 
Find the lengths of the diagonals. 

* When, directed by the instructor, check the solution of each triangle, 
t When using the law of sines without logarithms, we may introduce the cosecant 
in place of a sine which is in the denominator«if we desire to avoid performing 

c sin a 

division. Thus, we would write a = — - = c sin a. esc 7 . 

sin 7 








OBLIQUE TRIANGLES 127 

SUPPLEMENTARY PROBLEMS 

Solve by use of five-place logarithms: 


23. 

a = 21° 36.4'; 

0 = 82° 13.6'; 

a = 13.159. 

24. 

/3 = 48° 23.7'; 

y = 17° 16.3'; 

b = 27.001. 

25. 

0 = 23° 19.2'; 

y = 33° 40.8'; 

a = 513.52. 

26. 

a = 83° 45.1'; 

0 = 24° 18.9'; 

c = .75846. 

27. 

a = 19° 34.5'; 

0 = 80° 14.5'; 

c = .011105. 

28. 

a = 58° 11.4'; 

7 = 63° 25.6'; 

b = .0031478. 


For any triangle ABC, write the specified formula: 

29. For a, in terms of b, a, and 0. 31. For b, in terms of a, a, and 0. 

30. For c, in terms of a, a, and y. 32. For a, in terms of c, a, and y. 

33. Without substituting special values for letters, write an outline 
for the logarithmic solution of triangle ABC if we are given b, 0, and y. 

34. Prove that by use of figures like Figures 76 and 77. 

35. In triangle ABC, suppose that y = 90°. Show that in this case the 
equations of the law of sines give merely the well-known expressions for the 
sines of the two acute angles of the right triangle ABC. 

92. Case II, the ambiguous case; given two sides and an angle 
opposite one of them. We shall find that there may exist two solutions, 
or just one solution, or no solution, depending on the given values. 
On account of the possibility that the data may lead to two triangles, 
Case II is called the ambiguous case. 

If the given parts are a, b, and a, any triangle ABC satisfying the 
data may be constructed as follows (see page 128). 

1. Construct a with one side AD horizontal. On the other side of a measure 
off b from A to find C. 

2. With C as a center and a as a radius, strike an arc of a circle. The 
vertex B may be placed wherever this arc cuts AD. 

In Figure 78 or Figure 83, the arc does not meet AD, and hence 
there is no solution. In Figure 79, there is just one position for B, 
with (3 = 90°. In Figure 80, the arc cuts AD at B and Bi; hence 
there are two solutions, triangles ABC and ABAC. In Figure 81 
or Figure 82, there is just one solution. 



128 


TRIGONOMETRY 




B D 





Fig. 83 


★Note 1 . Let h be the perpendicular distance from C to AB ; then h — b sin a. 
In Figures 78 to 80, where a < b and a < 90°, we have no solution, just one, or 
two solutions according as a, the radius of the arc, is less than, equal to, or greater 
than h. These remarks and Figures 81 to 83 justify the following summary, 
if a, b, and a are given: 

(1) a < b sin a; no solution (Figure 78). 

( 2 ) a = b sin or, just one solution, with j 8 = 90°.(Figure 79). 

(3) a> b sin a, but a <b; two solutions (Figure 80). 

(4) a~2ib; just one solution (Figure 81). 


a < 90° 


a >90° 


(1) a>b\ just one solution (Figure 82). 

(2) a ^ &; no solution (Figure 83). 


93. Solution of Case II. Let a, b, and a be the given parts. 

1. Construct the triangle to scale, as a rough check. 

2. Find (3 by use of sin /3 = ; if this gives 

(i) sin j8 > 1, or log sin /3 > 0, then there is no solution; 

(it) sin /3 = 1, or log sin /3 = 0, then = 90°; just one solution; 

(m) sin /3 < 1, then find one acute and one obtuse value for /3; 
next step decides which, if any, of these values may be used. 

3. For each value of /3, compute y by use of y = 180° — (a + /3). Discard 
any value of (3 for which (a + | 8 ) ^ 180°, because y must be positive. 

a sin y 


the 


4. For each pair of values of (3 and y, compute c = 


sin a 












OBLIQUE TRIANGLES 


129 


Example 1. Solve triangle ABC if a = 20 , b = 10 , and a = 30°. 
Solution. A figure for this problem would have the general nature of Figure 81. 


Formulas 

Computation 


Data: a = 20; b = 10; a = 30°. 

sin B sin a 

— 7 — = - > or 

b a 

. 0 b sin a 

sm p = - 

a 

. a 10 sin 30° 
sm 0 =- 20 - = -2500. 

$ = 14° 29'. (From Table VII) 


Comment. We also obtain (3 = 180° — 14° 29', or 
0 = 165° 31'. Figure 81 shows conclusively that this value 
cannot be used because there is only one solution, with 
/3 acute. To prove this otherwise, we compute 
a + (3 = 30° + 165° 3U = 195° 31'; 
since this is > 180°, hence /3 = 165° 31' is impossible. 

7 = 180° - (a + 0 ). 

7 = 180° - (30° + 14° 29') = 135° 31'. 

c a 

—. - - —- > or 

sm 7 sm a 

a sin 7 

c = — -- • 

sm a 

sin 135° 31' = sin (180° - 135° 31') = sin 44° 29'. 

20 sin 44° 29' OQ AO /rr , U1 TrTTN 

c = -• — S 7 T 5 — = 28.03. (Table VII) 

sm 30 


Summary. One solution: c = 28.03; /3 = 14° 29'; 7 = 135° 3U. 


Example 2 . Solve triangle ABC if a — 
Solution without logarithms. sin 0 


2, b = 6 , and a = 30°. 
_ b sin a _ 6 (f) _ , _ 

" ~ET ~ ~ 15; 


this is greater than 1. Hence, there is no solution because the sine of an angle 
is never greater than 1. A figure for this problem would look like Figure 78. 

log b = 0.7782 

log sin a — 9.6990 — 10 ( + ) 
log b sin a = 0.4772 

log a = 0.3010 ( —) 

log sin j 8 = 0.1762, which is positive. 

If the logarithm of a number is positive, the number is greater than 1. Hence, if 
log sin 0 = 0.1762 then sin 0 > 1, which is impossible. Therefore, there is no 
solution. 


Solution with logarithms. 

. a b sin a 

sin 0 = -: 

a 


Example 3. How many triangles exist with a — 4, 6 = 8 , and a — 30°? 
Solution. 1. A figure for this problem would look like Figure 79. 

b sin a 8 sin 30° 


2. To find 0: 


sin 0 = 


80) 

4 


a 4 

sin /3 = 1, and /3 = 90°. There is just one solution, a right triangle. 


Hence, 





























130 


TRIGONOMETRY 


★Note 1 . Each solution of a problem under 
Case II may be checked by U3e of one of the pro¬ 
jection formulas 12 of page 122. Thus, if a, b, and 
a are the given parts of triangle ABC, each solu¬ 
tion may be checked by use of c = a cos 0 + b cos a. 

Example 4. Solve triangle ABC if 
b = 4.157; c = 3.446; 7 = 51° 48'. 

Solution. From Figure 84, it appears that 
there are two solutions, triangles ABC and ABiC, 
in which 0 = 70° and 0i = 110°, approximately. 


A 



Formulas 

Computation 


Data: b = 4.157; c = 3.446; 7 = 51° 48'. 

sin 0 sin 7 

b c ’ 

or 

. 0 b sin 7 

sin 0 -- 

c 

log b = 0.6188 (Table V) 

log sin 7 = 9.8953 - 10 ( + ) (Table VI) 

log b sin 7 = 10.5141 — 10 

log c = 0.5373 (-) 

log sin 0 = 9.9768 - 10; 0 = 71° 26'. (Table VI) 
Hence, 0! = 180° - 0 = 108° 34'. 


Solution for triangle ABC 

a = 180° - (0 + 7 )- 

a = 180° - (51° 48' + 71° 26') = 56° 46'. 

a c 

sin a sin 7 

or 

c sin a 

log c = 0.5373 
log sin a = 9.9224 — 10 (4-) 
log c sin a = 10.4597 — 10 
log sin 7 = 9.8953 — 10 ( —) 

Q — . 

sin y 

log 0 = 0.5644; a = 3.668. 


Solution for triangle AB\C 

a l = 180° - (/3i + 7 )- 

ai = 180° - (108° 34' + 51° 48') = 19° 38'. 

c sin a 1 
a 1 = - ;- 

sin 7 

log c = 0.5373 

log sin a 1 = 9.5263 — 10 (+) 
log c sin e*i = 10.0636 — 10 
log sin 7 = 9.8953 — 10 ( —) 

log a 1 = 0.1683; a x = 1.473. 


Summary. First solution-. a = 3.668; a = 56° 46'; 0 = 71° 26'. 

Second solution: a = 1.473; oci — 19° 38'; 0i = 108° 34'. 


Check for first solution, c = a cos 0 + b cos a.. 

a cos 0 = 3.668 cos 71° 26' = 1.168 (Table VII) 

b cos a = 4.157 cos 56° 46' = 2,278 (+) (Table VII) 

a cos 0 b cos a = 3.446; satisfactory. 


c = 3.446 




































OBLIQUE TRIANGLES 


131 


EXERCISE 52 

Solve each triangle without logarithms by use of Table VII: 


1-/3 = 30°; a = 4; 6 = 5 . 

2. T = 22° 20'; a = 50; c = 38. 

3. 7 = 65° 30'; 6 = 97i; c = 91. 

4 . a = 30°; 6 = 3; a = 5. 

5. 7 - 75°; c = 7; 6 = 7. 

6. a = 150°; a = 5; c = 8 . 

7. j 8 = 67°; a = 12; 6 = 6 . 


8 . a = 31° 20'; 6 = .25; a = .13. 

9. a = 157° 40'; a = 38; 6 = 25. 

10. a = 15°; c = 17; a = 17. 

11. 0 = 114° 30'; 6 = 13; c = 10 . 

12 . 7 = 148° 40'; a = 20; c = 26. 

13. 7 = 54°; a = 5; c = 2. 

14. 0 = 22° 20'; a = 50; 6 = 19. 


*Sofoe 6 y use of four-place or five-place logarithms; check each triangle 
when directed by the instructor: 


15. 

0 

= 

42° 

30'; 

6 

= 

16.7; c = 12.3. 

16. 

a 

— 

67° 

40'; 

a 

= 

2.39; c = 1.67. 

17. 

7 

= 

53° 

20 '; 

6 

= 

265; c = 241. 

18. 

0 

•«* 

24° 

10 '; 

a 

= 

136; 6 = 112. 

19. 

a 

= 

76° 

19'; 

a 

= 

.0572; 6 = .139. 

20 . 

a 

= 

71° 

45'; 

a 

- 

.9632; 6 = .9632. 

21 . 

0 

= 

36° 

53'; 

c 

= 

.07531; 6 = .05126. 

22 . 

7 

= 

49° 

46'; 

6 

m, 

.9652; c = .4738. 

23. 

a 

= 

113 

° 20 ' 

; a = 

= 11.56; c = 7.282. 

24. 

7 

- 

98° 

40'; 

a 

= 

.6495; c = .8307. 

25. 

0 

« 

47° 

26'; 

a 

= 

1356; b = 1356. 

26. 

a 

= 

65° 

34'; 

a 

= 

10.73; c = 9.835. 

27. 

7 

m 

147 

° 5'; 

6 

= 

19.36; c = 15.17. 

28. 

0 

= 

59° 

13'; 

a 

= 

28.43; b = 25.98. 

29. 

a 

= 

25° 

57'; 

a 

= 

1.278; b = 1.607. 

30. 

0 

= 

115 

° 37' 

; a = 

= .5216; b = .6735. 


Solve by use of five-place logarithms: 

31. a = 24° 7.4'; a = 1.2695; 6 = 1.5873. 

32. 7 = 36° 48.9'; c = 7.4261; 6 = 5.1379. 

33. a = 41° 9.2'; a = 16631; c = 12395. 

34 . 0 = 44 ° 13.3'; c = 6.6260; 6 = 4.6212. 

35. 0 = 52° 19.1'; 6 = 2.4017; c = 2.6317. 

36. a = 62° 4.8'; a = 2.3814; c = 2.6492. 

37. a = 35° 28'; a = 9.8470; 6 = 5.7135. 

38. 7 = 127° 19'; a = .74215; c = .97316. 

39. A telegraph pole is supported by two guy wires which run from 
the top of the pole to the ground on opposite sides. One wire is 53.6 feet 
long and makes an angle of 65° 20' with the ground. The other wire is 
57.5 feet long. Find the acute angle which the second wire makes with 
the ground, if both wires are in the same vertical plane. 


132 


TRIGONOMETRY 


44. 

0 

= 115° 

; a — 35; 

b = 56. 

45. 

7 

= 109° 

; a = 170: 

; c = 59. 

46. 

7 

= 41°; 

a = 200 ; 

c = 135. 

47. 

a 

= 35°; 

a = 18; c 

: = 30. 


SUPPLEMENTARY PROBLEMS 

Without solving the triangle, apply the tests of Note 1, page 128, to deter¬ 
mine how many solutions exist: 

40. a = 58°; a = 42; b = 50. 

41. 0 = 27°; b = 19; c = 30. 

42. 7 = 73°; a = 60; c = 58. 

43. t - 62°; 6 = 50; c = 38. 

48. A horizontal force of 85 pounds pulls due east on an object at P. 
A second horizontal force pulls on P in a direction 37° west of north. 
The magnitude of the resultant of these forces is 73 pounds. Find the 
magnitude of the second force and the direction of the resultant. 

94. The law of tangents. In any triangle, the difference of any 
two sides, divided by their sum, equals the tangent of one half of the 
difference of the opposite angles divided by the tangent of one half of 
their sum: 

a — b _ tan \{a — 1 8 ) c — a _ tan 3(7 — a) 

a + b ~ tan \{ot -f- 0) ’ c + a tan §{7 + a) ’ 

b - c tan J(0 — 7) 


b + c tan |(0 + T) 

Proof. 1. Let b and c be any two sides and let us prove (2). 

2. By the law of sines, - = S | D 

c sin 7 

3. Subtract 1 on both sides of (3); also, add 1 on both sides: 


c ^ 


( 1 ) 

( 2 ) 


c +1 = 


sm 0 . 

-—- — 1 ; or 

b - c 

sin 0 — sin 7 

sm 7 

c 

sin 7 

sin 0 

■ + i; or 

b + c 

sin 0 + sin 7 

sm 7 

c 

sin 7 


(3) 


(4) 


(5) 


4. Divide each side of (4) by the corresponding side of (5); then use 
formulas XXI and XXII from page 111: 

b — c _ sin 0 — sin 7 _ 2 sin |(0 — 7 ) cos |(0 + 7 ) 
b + c sin 0 + sin 7 2 cos ^(0 — 7 ) sin ^(0 + 7 ) 

Hence, ^ = tan M - 7 ) cot W + 7 ) - 

Note 1. Corresponding to each of (1) and (2), we may write an equivalent 
equation by changing the order of the letters. Thus, instead of (2), we may write 

c — b _ tan |(7 — 0 ) 
c + b tan |(7 + 0) 


(6) 






















OBLIQUE TRIANGLES 133 

95. Solution of Case III by use of the law of tangents. Let the 

given sides of triangle ABC be a , b , and y. 

1. Compute |(a: + 0) = 1(180° — 7 ). 

2. Find — 3) by use of the law of tangents. 

3. Compute a and 0 by use of results from Steps 1 and 2: 

« = Ka + 0) + - 0); 0 = l(a + 0) - l(a ~ 0). (1) 

4. Find c by use of the law of sines. 

5. Check by use of the law of sines. 


Note 1. In the following example, notice that, because c > b, we use formula 
6, page 132, instead of formula 2, page 132, to avoid negative differences. 

Example 1. Solve triangle ABC if a = 78° 48'; b = 726; c = 938. 


Formulas 

Computation 


Data: a = 78° 48'; b = 726; c = 938. 

1(7 + 0 ) = 1(180° - a). 

1(7 + 0) = 5(180° - 78° 48') = 50° 36'.- 

tan |(7 — 3) c — b 

= * or 

tan §(7 + 0 ) c + b’ ’ 

tan KT ~ 0) = 

(c - b) tan |(7 + 0) 
c + b 

log (c - b) = 2.3263 
log tan K7 + 0 ) = 0.0855 (+) 

c = 938 
b = 726 
c-b = 212 
. c + b = 1664. 

log numerator = 12.4118 — 10 
log (c + b) = 3.2211 (-) 

log tan 1(7 — 0) = 9.1907 — 10 
1(7 - 0 ) = 8 ° 49'. 

(Table VI) 

0 = Kr + 0) - 1(7 - 0). 

7 = 1(7 + 0) + 1(7 - 0)- 

1(7 + 0) = 50° 36' \ Add to get 7 . 

2(7 — 0) = 8 ° 49' / Subtract to get 0. 

7 = 59° 25'; 0 = 41° 47'. 

0 c 

sin a sin 7 ’ 

or 

c sin a 

log c = 2.9722 
log sin a = 9.9916 — 10 (+) 
log c sin a = 12.9638 — 10 
log sin 7 = 9.9350 — 10 ( —) 

a — 

sin y 

log a = 3.0288; a = 1069. 


Summary. 0 = 41° 47'; 7 = 59° 25'; a = 1069. 


Check. 


a 

sin a 


b 


sin 0 


log a = 13.0289 - 10 
log sin a — 9.9916 — 10 ( —) 

log a/sin a = 3.0373-* 


log b = 12.8609 - 10 
log sin 0= 9.8237 - 10 (-) 
log 6/sin 0 = 3.0372; satisfactory. 


Comment. Notice that we could have used as a check formula any one of the 
formulas of the law of tangents which was not used in the solution. 
































134 


TRIGONOMETRY 


96. Use of the law of tangents to check in Case I and Case II. 

Example 1. Check the first solution in Example 4, page 130. 

Solution. The data were b = 4.157, c = 3.446, and 7 = 51° 48'. The 
solution was a = 3.668, a = 56° 46', and /3 = 71° 26'. To check, we use 

b — a tan HP — a) 


b + a tan 5 ()3 + a) 


b = 4.157 
a = 3.668 


/3 = 71° 26' 
a = 56° 46' 


b + a = 7.825 $ + a = 128° 12' 

b — a = .489 /3 — a = 14° 40' 

Hi3 + a) = 64° 6'; §(/3 - a) = 7° 20'. 

/ log (5 - a) = 9.6893 - 10 J log tan |(/3 - a) = 9.1096 - 10 

\ log (b + a) = 0.8935 (-) \ log tan HP + a) = 0.3137 (-) 


log ? = 8.7958 - 10- 

O + d 


log tan S(/3 - = g 7Q59 _ 1Q 

& tan HP + «) 


★97. Mollweide’s equations. In any triangle ABC, 

a - b _ sin \{a - ff). b - c _ sin ^(fl - 7) . c - a = sin §(7 - a) , 

c ~ cos ^7 0 cos b cos §0 

a + b _ cos \{ot - ft), b +_c _ cos §Q3 - 7 ) . a + c _ cos £(7 - <*) 

c ~ sin Jy ’ a sin \a ’ b sin J/3 

Proof. 1. Let us prove the middle equation in (1). 


5. Use (7), and (XXII) of page 111, in (5): 
b — c 2 cos HP + 7 ) sin HP ~ 7 ) 


sin HP ~ 7) 


( 1 ) 

( 2 ) 


2. From equation 4, page 132, 

b - c 

c 

(3) 

sin /3 — sin 7 

sin 7 

a- c - 0 

b — c 

a 

(4) 

foince . > 

sm 7 sm a 

sin P — sin 7 

sin a 

b — c sin P — sin 7 
Hence. — 


(5) 

' a 

3. By use of (IX), page 107, 

sin a 

sin a = 2 sin \a cos \ot. 

( 6 ) 

4. Since a = 180° — (/3 + 7 ), 

\a = 90° - 

HP + 7 ). 


Hence, sin %a = cos HP + 7 ), 

sin a = 2 cos ^ 

and, from ( 6 ), 

( 1 8 + 7 ) cos \a. 


(7) 


a 2 cos HP + 7 ) cos %a cos 

This establishes equations 1. Equations 2 may be proved similarly. 

Note 1. Mollweide’s equations are named after a German astronomer, 
Karl Mollweide (1774-1825), but the equations were known long before his 
time. These equations are useful in checking solutions of a triangle because each 
equation involves all parts of the triangle. 


































OBLIQUE TRIANGLES 


135 


EXERCISE 53 

Solve by use of four-place or five-place logarithms and check:' 


1. b = 387; c = 136; a = 68 °. 

2. b = 2.64; c = 1.07; a = 56°. 

3. a = 102; b = 437; y = 43°. 

4. a = .936; c = .348; 0 = 72°. 

5. b = .657; c = .319; a = 108°. 

6 . a = 3045; c = 1726; /3 = 47°20'. 


7. b = 13.16; c = 22.78; a = 69° 40'. 

8 . a = 773.6; c = 993.4; yS = 120° 30'. 

9. c - .01392; a = .00756; /3 = 43° 28'. 

10. b = 2.731; c = 3.914; a m 98° 24'. 

11. a = .936; c = 2.578; 0 = 23° 17'. 

12. a = 21.45; b = 9.36; y = 70° 25'. 


Solve by use of five-place logarithms and check: 


13. a = 21.467; 

14. b = 316.25; 

15. a = .98315; 

16. b = .84107; 

17. a = 31024; 


6 = 13.218; 7 = 19° 20.6'. 

c = 158.67; a = 61° 19.8'. 

c = 1.25670; 0 = 57° 35.7'. 

c = 1.36450; a = 98° 16.8'. 

c « 65937; 0 = 58° 7.4'. 


18. b = 13056.0; c - 8947.2; a = 105° 6.5'. 


SUPPLEMENTARY PROBLEMS 

Solve by use of the law of tangents, without logarithms; use Table VII: 

19. a = 12; 6 = 8 ; y — 30°. 22. b = 30; c = 25; a = 50°. 

20. a = 10; b = 15; y = 150°. 23. a - 50; c = 25; 0 = 110°. 

21. a = 10; 6 = 20; y = 40°. 24. a = 20; c = 40; 0 = 38°. 

Solve by use of four-place or five-place logarithms, and check by use of 

the law of tangents: 

25. a = 163; a = 27° 20'; 0 = 16° 57'. 

26. c = 21.36; a = 62° 15'; y = 21° 37'. 

27. b = .1327; a = 63° 17'; 0 = 42° 21'. 

28. a = .367; b = .458; 0 = 39° 24'. 

29. a = 431.6; c = 693.2; y = 68 ° 29'. 

30. b = 385; c - 447; 0 = 41° 28'. 

31. The sides of a parallelogram are 185 feet and 263 feet long, and 
one angle is 39°. Find the length of the longest diagonal. 

32. Two forces whose magnitudes are 340.6 pounds and 263.5 pounds 
act simultaneously on an object. If the angle between the directions of 
the forces is 63° 48', find the magnitude of the resultant force. 

33. Prove the middle equation in formulas 2 , page 134. 

34. Prove the first equation in formulas 1, page 132. 

* The instructor may desire to require some use of Mollweide’s equations. 


136 


TRIGONOMETRY 


98. The inscribed circle. We recall that the bisectors of the angles 
of a triangle meet at the center of the inscribed circle. Let r be its 
radius; then, in Figure 85, 

. OD = OE = OF = r. 

Let s be one half of the perimeter of 
[E triangle ABC: 

s = *(a + b + c). (1) 

Let K be the area of triangle ABC: 



and 


Fig. 85 


B Then, we shall prove that 
K = rs, 


•=v 


(s - a)(s - b)(s - c) 

s 


( 2 ) 

(3) 


Proof. 1. The area of triangle BOC is a OE ■ BC = \ra. 

Similarly we find the areas of triangles AOC and AOB. 

2. K = (area of BOC ) + (area of AOC ) + {area of AOB). 
Hence, K = \ra + \rb + %rc = r ■ \{a + b + c) = rs. 

3. In geometry, and also later in this chapter, it is proved that 


K = Vs{s — a){s — b){s - c). 


(4) 


4. From (2) and (4), rs = Vs(s — a)(s — b)(s — c ). 

On dividing both sides of this equation by s, or Vs®, we obtain (3). 

99. Tangents of the half-angles. In any triangle ABC, 
r . 0 r Y r 


a 

tan — = 

2 s — a 


tan ^ = 

2 s — b’ 


tan ^ 

2 s — c 


( 1 ) 


( 2 ) 


Proof. 1. Let a represent any angle of the triangle. Then, from 

r 

AD 

2. The whole perimeter is 

2s = (AD + AF) + (BE + BD) + {CE + CF). 


triangle ADO in Figure 85, 


a OD 
tan - = = 
2 AD 


But, from Figure 85, AF = AD; BD = BE; CF = CE. Therefore, 
2s = 2AD + 2 BE + 2 CE; or s = AD + (BE + CE). 

Since BE + (JE = BC = a, then s = AD + a; or, AD = s — a. (3) 

Oi T 

3. From (3) and (2), tan — -- Hence, the tangent of one half of 

A 5 — a 

any angle equals r divided by the difference of s and the opposite side. 












OBLIQUE TRIANGLES 


137 


100. Functions of the half-angles.* 

triangle ABC; then we shall prove that 


Let a be any angle of 


sin 


s-V 


(s - b)(s - c) 

be 


cos 


l-V 


s(s — a) 
be 


tan - = 


( 1 ) 


Proof. 1. The following useful equations are easily verified by sub¬ 


stituting s = 5 (° + & + c )- 
a + b + c = 2s; 
a — b + c = 2 (s — 6); 

2. From formulas XVI, page 107, 

2 cos 2 ia = 1 + cos a; 

3. By the law of cosines, 


6 

+ C 

— a = 2(s 

- a); 

(2) 

a 

+ 6' 

1 

II 

to 

/ 5 N 

- c). 

( 3 ) 

2 

sin 2 

= 1 — 

cos a. 

( 4 ) 



6 2 + c 2 — 

a 2 

( 5 ) 

cos 

a = 

26c 



n- o , « 1 , 6 2 + c 2 — a 2 (6 2 + 26c + c 2 ) — a 2 

Hence, 2 «*■ 2 " 1 +- Wc - =-2ite- ; 

, a _ (6 + c) 2 — a 2 (6 + c + a) (6 + c — a) 

C0S 2 ~ 46c 46c 


Use (2) in (6): cos 2 ^ or 


cos 




s(s — a) 
be 


4. Similarly, 


sim -s = 


„ . , a , 6 2 + c 2 - a 2 a 2 - (6 - c) 2 

2 SIn 2 = 1 - Wc - = - Wc - ; 

(a — b + c)(a + b — c) _ (s — b)(s — c). 


46c 


6c 


or 


. a 

sm 2 




(* - b)(s - c) 
be 


( 6 ) 

( 7 ) 


( 8 ) 


5. By use of (7) and (8), 


tan \a 


sin 

cos 


/ ( 8 - fr)(s - c) 

\ s(s - a) 


Multiply numerator and denominator of the radicand by (s — a): 

a l(s - a)(s - b)(s - c ) 1 _ /(« - a)(* - 6)(s - c) /m 

tan 2“ V-«(. - «)*'- ~T^\ r - 

If we let r = ~ a K* - then tan ^ - - j ;—• 

Note 1. In the preceding derivation, r is introduced without any geometrical 
interpretation, merely as an abbreviation for a complicated radical. Hence, 
Section 98 is not needed for deriving (1). Nevertheless, Section 98 is useful in 
showing the geometrical interpretation of r. 

* The instructor may desire to omit this section in a brief course. 
































138 


TRIGONOMETRY 


Note 2. By altering the letters in (1) on page 137 we obtain 


dn I - V 
sin 1 - V 


(s — g)(s — c). 
oc 


(s — a)(s — 6 ) 
ab 


■w 

V 


s(s — b). 
oc ’ 


s(s — c) 
ab ’ 


/3 _ r 
tan 2 ~ s - 6 ’ 


y r 

tan f- -- 

2 s — c 


(10) 

( 11 ) 


101. Solution of Case IV by the half-angle formulas. 

1. Compute s, (s - a), (s - b), (s - c), and then log r. 

2. Find (a, /3, 7 ) by use of the tangents of the half-angles. 

3. Check by use of a + /3 + 7 = 180°. 


Example 1. Solve triangle ABC if a — 173; b = 267; c = 412. 
Solution. We use formulas 1 of Section 99. 


Formulas 

Computation 

s = i(a + b + c). 

To check here, notice that 
(s — a) + (s — b) + (s — c ) 

= 3s — (0 + b + c) = s. 

a = 173 ] s - a = 253 ] 

b = 267 l (+) s - b = 159 1 (+) 

c = 412 J s — c = 14 1 

2s - 852-^ <-s = 426 


log (s - o) = 2.4031 1 

I(s - o)(s - b) (s — c) # 

log (s - b) = 2.2014 i (+) 

r “ \ s 

. (s - a)(s - b)(s - c ) 
s 

log r = 5 log r 2 . 

log (s - c) ■= 1.1461 j 

log numerator = 5.7506 

log s = 2.6294 (-) 

log r 2 = 3.1212; (- 5 - by 2) 
log r = 1.5606. 

a r 

log r = 11.5606 - 10 
log (s - 0 ) = 2.4031 (—) 

tan 2 - s - a 

log tan %a = 9.1575 — 10; 

ia = 8 ° 11 '; a = 16° 22 '. 

tan P = r • 

log r = 11.5606 - 10 
log (s - 6 ) = 2.2014 (-) 

2 s — b 

log tan = 9.3592 - 10; 

4/8 = 12° 53'; /3 = 25° 46'. 

ta „T_ r 

log r = 1.5606 
log (s — c) = 1.1461 (—) 

ian — 

2 s — c 

log tan £7 = 0.4145; 

|7 = 68 ° 56'; 7 = 137° 52'. 

Summary. 

a = 16° 22'; |8 = 25° 46'; 7 = 137° 52'. 

Check, a + p + 7 = 16° 22 

!' + 25° 46' + 137° 52' = 180° O'; satisfactory. 



































OBLIQUE TRIANGLES 


139 


EXERCISE 54 


Solve by use of four-place or five-place logarithms and check: * 


1. 

0 = 17; b 

= 26; c 

= 35. 

10 . 

a = 128.3; 

b = 

140.7; 

c = 188.4. 

2 . 

a = 12; b 

= 23; c 

= 18. 

11 . 

a = 4.614; 

b = 

6.213; 

c = 5.709. 

3. 

a = 8; b = 

-- 7; c = 

11 . 

12 . 

a — 4.908; 

b = 

3.652; 

c = 6.404. 

4. 

a = 23; b 

= 37; c 

= 48. 

13. 

a = 1563; 

b = 

1198; < 

3 = 2206. 

5. 

a = 5.26; b 

= 4.38; 

c = 9.34. 

14. 

a = .3158; 

b - 

.2893; 

c = .1642. 

6 . 

a = .986; b 

= .726; 

c = .648. 

15. 

a = .1931; 

b = 

.1137; 

c = .2625. 

7. 

a = .342; b 

= .597; 

c = .651. 

16. 

a = .06534 

)b = 

.07200 

; c = .04099. 

8 . 

a = 136; b 

= 472; 

c = 450. 

17. 

a = 73.09; 

b = 

91.27; 

c = 59.86. 

9. 

a = 19; b ■ 

= 21 ; c 

= 35. 

18. 

a = 5.936; 

b = 

9.103; 

c = 4.967. 


Solve by use of five-place logarithms and check: 

19. a = 12.379; b = 26.423; c = 25.603. 

20. a = 3812.6; b = 5372.4; c = 7635.8. 

21. a = .91315; b = .80263; c = .74137. 

22. a = .072130; b = .057387; c = .043002. 

23. a = 9.8314; b = 6.1041; c = 7.0308. 

24. a = 111.78; b = 235.67; c = 204.38. 


SUPPLEMENTARY PROBLEMS 

Note. If we desire to find only one angle in a triangle whose sides are given, 
usually it is most convenient to use the formula for the sine, or the cosine, of the 
half-angle instead of the formula for its tangent. Use this method below. 


Find only the specified angle by use of logarithms: 


25. a = 13; b = 17; c = 9: find /3. 29. Find/3 in Problem 14. 

26. a = 27; b = 36; c = 42: find 7 . 30. Find 7 in Problem 15. 


27. a = 73.1; b = 26.7; c = 83.5: find a. 31. Find a in Problem 19- 

28. a = 113.1; b = 207.5; c = 315.6: find a. 32. Find /3 in Problem 20. 

33. Two forces, of 132.5 pounds and 217.3 pounds, act simultaneously 
on an object. The magnitude of the resultant force is 265.3 pounds. Find 
the angle between the directions in which the given forces act. 

34. Three circles are tangent to each other externally, and their radii are, 
respectively, 175 inches, 236 inches, and 182 inches. Find the angles of the 
triangle formed by the lines joining the centers of the circles. 


* In checking, it is not unusual to find that the sum of the angles differs from 
180° by several units in the last angular place being used. This is due to the fact 
that each whole angle is subject to twice the unavoidable error which may rise in 
computing the half-angle. 


TRIGONOMETRY 


140 

102. The area of a triangle can be expressed conveniently in terms 
of the given parts of the triangle under each of the following cases. 

Case I. Given one side and two angles. 

Case III. Given two sides and the included angle. 

Case IV. Given three sides. 

The area in terms of two sides and the included angle: 

K = \bc sin a; K= \ac sin /3; K - \ab sin y. 

Proof. Let a and c be any two sides, and let h A 

be the altitude from A to BC, in Figure 86. 

Then, regardless of the size * of /3, 
h = c sin /3. 

Hence, K = \ha = \ac sin (3. 


angles: 


a 2 sin/3 sin y. ^ b 2 sin a sin y v _ c 2 sin a sin (3 
K= 2 sin a ’ K ~ 2 sin p ’ 2 sin y 


Proof. 1. From (1), 

2. From the law of sines, 


K = Jac sin 
sin y 


sin a 


-y or 


c = 


a sin y 
sin a 


__ . a sin y . 0 a 2 sin y sin /3 

K = \a — . - - sin p = — 0 — ;- 

2 ^ 2 sin a 


sin a 


43.3. 


„ . a a 
sin a = 2 sin cos ' 


„ , . a a 

K = be sm ^ cos 


( 1 ) 



( 2 ) 

(3) 


(4) 


Hence, from (3), 

Example 1. Find the area if a = 10, /3 = 30°, and y = 120°. 

Solution, a = 180° - (j8 + y) = 30°. Hence, 

_ a 2 sin 0 sin y 100 sin 30° sin 120° = 100^3 = 

• ^ — 2 sin a 2 sin 30° 4 

The area in terms of the sides: 

K - Vs(s - a)(s — b)(s — c). 

Proof. 1. From (1), K = \bc sin a. 

2. By use of formula IX, page 107, 

Hence, 

3. By use of formulas 1, page 137, 

* For the case where /3 is obtuse, see Figure 77 and proof on page 124 for suggestion. 




















OBLIQUE TRIANGLES 


141 


EXERCISE 55 

Find the area of the triangle without using logarithms: 

1. a — 10; c = 8 ; 0 = 30°. 7. a = 10; cc = 15°; 0 = 150°. 

2 . b = 2; c — V 3 ; a = 60°. 8. & = 20 ; 0 = 22 ° 30'; 7 = 135°. 

3. a = 3; b = V§> 7 = 120 °. 9. a = 7; b = 12 ; c = 13. 

4. b = 15; c = 10; a = 150°. 10. a = 9; & = 17; c = 10. 

5. a = 36°; 6 = 15; c — 2. 11. a = 21; 6 = 9; c = 20. 

6 . 7 = 105°; a = 20; 6 = 30. 12. a = 11; 6 = 18; c = 25. 

Compute the area by use of four-place or five-place logarithms: 

13. a = 4.908; 6 = 3.578; c = 7.404. 


14. a 

15. a 

16. a 

17. c 

18. 6 

19. c 

20 . a 

21. C : 

= 128.3; 6 = 140.7; c = 188.4. 

= .342; 6 = .726; c = .648. 

= .3158; 6 = .2893; c = .1642. 

= 31.42; a = 37° 31'; 0 = 63° 45'. 

= .0157; a = 62° 17'; 0 = 53° 25'. 

= .1963; a = 51° 45'; 0 = 62° 17'. 

= 151.3; 6 = 206.3; 7 = 42° 17'. 

= .9135; 6 = .8646; a = 85° 14'. 


irFind the area of the inscribed circle in the specified problem: 

22. Problem 9. 23. Problem 10. 24. Problem 13. 25. Problem 14, 

103. Summary concerning the solution of triangles. 


I. Given two angles 
and a side. 

Solve by law of sines and check by law of tangents* 

II. Given two sides 
and an opposite angle. 

Solve by law of sines, with particular attention to 
the number of solutions. Check by law of tangents* 

III. Given two sides 
and the included angle. 

(1) Find the angles by law of tangents and the 
third side by law of sines. Check by law of sines 
or law of tangents* 

(2) If only the third side is desired, perhaps use 
law of cosines (not adapted to logarithms). 

IV. Given three sides. 

(1) Solve by half-angle formulas and check by 
a + 0 + 7 - 180°. 

(2) Solve by law of cosines (not adapted to 
logarithms) and check by a + 0 + 7 = 180°. 


* Or, Mollweide’s equations; or, the formulas like c = a cos /3 + b cos a. 









142 


TRIGONOMETRY 


MISCELLANEOUS EXERCISE 56 

Solve each triangle completely except where otherwise specified. Use Table 
YII and Table I or Table XII, without logarithms. 

1. b = 20; 0 = 30°; 7 = 57°: find only side a. 

2 . a = V 3 ; a = 60°; 7 = 17°: find only side c. 

3. a = 3; b = 5; c = 7: find only 0. 11. b = 38; 0 = 22° 20'; a = 60. 

4. a = 6; b = 8; c = 5: find only a. 12. c == 40; /S = 90°; 7 = 26° 12'. 

5. a = 4; 6 = 5; c = 8 : find only 7 . 13. & = 81; c = 50; 0 = 164° 20'. 

6 . a = 20; b = 30; c = 15: find only 0. 14. a = 91; 6 = 40; a = 114° 30'. 

7. a = 20; a = 30°; 7 = 42° 7'. 15. c = 20; a = 50; a = 90°. 

8 . a = 5; 6 = 6 ; c = 3; find only 7 . 16. 6 = V2; c = 2; 0 = 45°. 

9 . a = 6 ; 6 = 4; c = 9; find only a. 17. a = 2000; 6 = 684; 0 = 20°. 

10. c = 27; 0 = 26° 12'; 7 = 15° 40'. 18. c = 27; 7 = 15° 40'; 0 = 50. 

Solve and check by use of four-place or five-place logarithms: 

19. a m 13.17; a = 26° 33'; 7 = 82° 58'. 

20. a = .956; 6 = .734; c = .526. 

21. a = 1.315; 6 = 2.673; 7 = 80° 19'. 

22. a = 395.6; 6 = 524.7; 0 = 73° 6 '. 

23. a = 160.7; c = 130.5; 7 = 47° 21'. 

24. 0 = 672.1; a = 27° 14'; 0 = 90° O'. 

25. 6 = .0385; c = .0629; 0 = 85° 8 '. 

26. 6 = 21.54; c = 42.36; a = 105° 16'. 

27. a = 863; 6 = 457; 0 = 121° 53'. 

28. 6 = 1573; c = 6132; 0 = 82° 14'. 

29. a = 31.48; 6 = 56.72; c = 40.09. 

30. 6 = 1498; 0 = 63° 18'; 7 = 15° 27'. 

31. a = 1.632; c = 2.537; 0 = 90° O'. 

32. a = 2395; c = 4647; a = 25° 17'. 

Note. Recall the suggestions on page 17 for outlining the solution of an ap¬ 
plied problem. 

33. The lengths of two sides and a diagonal of a parallelogram are, 
respectively, 30 feet, 40 feet, and 60 feet. Find the angles of the parallelo¬ 
gram without logarithms. 

34. An aviator finds that, after traveling 300 miles in an easterly 
direction and then 250 miles in a southerly direction, he is 400 miles due 
east of his starting point. Without logarithms, find the direction in which 
he traveled on the first leg of his journey. 


OBLIQUE TRIANGLES 


143 


Employ the methods of this chapter in connection with at least one oblique 
triangle in the solution of each of the following problems* Construct a figure 
and outline the solution in each problem before computing. 

35. A telegraph pole, which leans 15° 10' from the vertical toward the 
sun, casts a shadow 39.8 feet long when the angle of elevation of the 
sun is 63° 40'. Find the length of the pole. 

36. A flagpole stands on the ridge of a roof whose sides are inclined 
25° 20' from the horizontal. When the angle of elevation of the sun is 
73° 50', the pole casts a shadow 23.8 feet long directly down the roof. 
Find the height of the flagpole. 


37. At a point 3| miles from one end of a lake and 5f miles from the 
other end, the lake subtends an angle of 78° 53'. Find the length of the 
lake. 


38. Along one bank of a river with parallel 
banks (see Figure 87), a surveyor measures 
AB = 183.5 feet, ABAC = 67° 45', and 
Z ABC = 43° 26', where C is a point on the 
opposite bank. Find the width of the river. 


Outline. 

from 


1. In triangle ABC, find only log b 
^ _ c sin j3 
sin 7 


C 



2. From right triangle ADC, CD = b sin a. 


39. From a ship, the angle of elevation of a point B at the top of a 
cliff is 21° 13'. After the ship has sailed 2500 feet directly toward B, 
its angle of elevation is found to be 47° 17'. Find the height of the cliff. 

40. The angle at one corner of a triangular field is 38° 47'. The sides 
which meet at this corner are 457.7 feet and 365.4 feet long. Find the 
length of the other side of the field and its area in square yards. 

41. A playground slide is inclined 58° 20' from the horizontal and is 
18.6 feet long. A ladder 16.5 feet long reaches from the ground behind 
the slide to its top. At what angle is the ladder inclined from the hori¬ 
zontal? 


42. From the top of an inclined road, 256 feet long, a vertical tower 
at the bottom subtends an angle of 22° 43'. Find the height of the tower 
if the road is inclined 39° 50' from the horizontal. 


43. An aviator observes that the angles of depression of two points, A 
and B, due north of him, are 43° 50' and 35° 26'. If A and B are 3570 feet 
apart, in the same horizontal plane, find the elevation from which they 
were observed. 


* Answers are given in accordance with the agreement in Note 1, page 48. In some 
problems, the student may conveniently compute without logarithms. 







TRIGONOMETRY 


144 


44. The acute angle between the diagonals of a parallelogram is 27° 20' 
and the lengths of the diagonals are 30 feet and 20 feet. Find the sides 
of the parallelogram, without using logarithms of trigonometric functions. 

45 . At two stations, 8500 feet apart on a straight horizontal railroad, an 
airplane is observed over the railroad between the stations. At a certain 
instant, the angles of elevation of the airplane as seen from the stations 
are 42° 17' and 57° 18'. Find the elevation of the airplane. 

46. A triangular building lot at the intersection of two streets has a 
frontage of 267 feet on one street and 135 feet on the other. If the third 
side of the lot is 356 feet long, find the acute angle between the streets. 

47. Find the largest diameter which could be used for the circular 
base of a cylindrical oil tank to be placed on the lot in Problem 46. 

48. An automobile travels at a speed of 30 miles per hour along a hori¬ 
zontal road directly toward us. From a position on the top of a hill we 
observe that the angle of depression of the automobile is 33° 25', and 
then, one minute later, is 73° 50'. How high is the hilltop above the road? 

49. In traveling at a speed of 12 miles per hour down a straight river 
in flat country we observe a fire ahead of us which, at first, is 23° 40' to 
the left of our route, and then, five minutes later, is 67° 20' to the left. 
How far is the fire from the river? 

50. A tower, 75.8 feet high, stands at the top of an embankment which 
runs due east and west and slopes to the north. At noon, when the sun 
is 25° 33 ' south of the zenith, the tower casts a shadow 62.6 feet down the 
embankment. At what angle is the embankment inclined from the 
horizontal? Recall that the zenith is directly overhead. 


51. A surveyor runs a line AB = 

N 


Jir 

E 

vv 




Fia. 88 

N 47° 53' E. Find the length of 


1326 feet in the direction N 37° 28' E, 
then the line BC — 1184 feet in a 
southerly direction, and then the line 
CA — 1016 feet, back to his starting 
point. As a check, he measures the 
direction of CA. Find the direction 
he should obtain. 

52. A surveyor desires to prolong 
a straight line AB due east past an 
obstruction (see Figure 88 ). He 
measures BC = 785.4 feet, S 23° 17'E, 
and then he runs CD in the direction 
1 if D is to be due east of B. 


53. Find the circumference of the outer edge of the largest circular 
running track which could be installed in a triangular field whose sides 
are 357 feet, 342 feet, and 438 feet long. 






OBLIQUE TRIANGLES 


145 


54. An airfield A is due east of a second field B. At a certain instant, 
one airplane takes off from B at a speed of 135 miles per hour in the di¬ 
rection N 36° 50' E, and a second airplane takes off from A at a speed of 
110 miles per hour. Find the direction in which the second airplane 
travels if it meets the first at the end of 5 hours. 

55. A boat is towing an artillery target 2000 feet behind it. From the 
guns on the shore, the direction of the boat is N 38° 10' E and of the target 
is N 21° 43' E. Find the distance from the guns to the target, if the boat 

and the target are traveling due east. 

56. In Figure 89, to find the distance 

between the inaccessible points C and D, 
on one side of the river, a surveyor on the 
other side measures AB = 157.8 feet; 
8 = 32° 26'; 6 = 27° 45'; 0 = 40° 29'; 

7 = 35° 18'. Find CD. 

57. Two points, A and B, 25,500 yards 
apart, are directly below the path of an 

airplane which is flying at uniform speed in a straight line. At a certain 
instant, the angles of elevation of the airplane as observed at A and B 
are 20° 36' and 13° 25', respectively. Three minutes later the airplane is 
still between A and B and the angles of elevation are 9° 12' and 86° 17', 
respectively. Find the speed of the airplane in miles per hour. 

58. To find the distance between two points A and D on opposite sides 
of a forest, without entering it, a surveyor runs the lines AB = 2725 yards, 
BC = 2246 yards, and CD = 3629 yards, in the directions N 26° 41' E, 
N 78° 18' E, and S 21° 34' E, respectively. Find AD and the direction 
of D from A. 

59. From the top of a hill, 326 feet above a lake, the angles of depression 
of the ends of the lake are 37° 20' and 42° 36', and a line joining the ends 
of the lake subtends an angle of 103° 28'. Find the length of the lake. 

60. Two towers, A and B, on the shore of a 
lake can be observed from only one point C on 
the opposite shore. The line joining the bases 
of the towers subtends an angle of 63° 42' at C. 

The heights of the towers are 132 feet and 89 
feet, and the angles of elevation of their tops as 
seen from C are 8° 13' and 7° 21', respectively. 

Find the distance AB. 

61. To find the distance between A and D, 
at opposite ends of a lake (see Figure 90), we 
measure AB — 186 yards, BC = 216 yards, 

BD = 242 yards, BE = 168 yards, and 
EC = 264 yards. Find the distance AD. 




Fig. 90 








146 


TRIGONOMETRY 


PROBLEMS INVOLVING FORCES AND VECTORS 

Note. In any problem, all forces or vectors will be supposed to lie in the same 
plane. Forces or vectors are directed horizontally, unless otherwise stated. 

Find the magnitude and the direction of the resultant force if the specified 
forces act together on an object. 

62. 585 pounds directed N 26° 25' E ; 243 pounds directed N 78° 47' W. 

63. 41.7 pounds directed S 17° 5' E; 39.4 pounds directed N 21° 14' E. 

64. 97.3 pounds directed N 29° 17' W ; 53.4 pounds directed S 73° 45' E. 

65. 705.4 pounds directed S 10° 22 ' W ; 918 pounds directed N 73° 44' E. 

66 . 175 pounds directed due east; 368 pounds directed upward at an 
angle of 17° 36' due east from the vertical. 

67. 267 pounds directed due north; 653 pounds directed upward at an 
angle of 23° 17' due south from the vertical. 

68 . One force of 8 pounds is directed due east. The magnitude of a 
second force is 12 pounds, and the magnitude of the resultant of these two 
forces is 15 pounds. Find the direction of the second force if it acts some¬ 
what east of north, and find the direction of the resultant. 

69. One force of 15 pounds is directed due west. The magnitude of 
a second force is 25 pounds and the magnitude of the resultant of these two 
forces is 30 pounds. Find the direction of the second force, if it acts 
somewhat west of south, and find the direction of the resultant. 

70. Find the sum of two vectors whose magnitudes are 53.17 and 21.68, 
and whose directions are N 37° 20' E and S 67° 25' E, respectively. 

71. Find the sum of two vectors whose magnitudes are 131.4 and 781.5, 
and whose directions are N 26° 15' W and S 78° 17' E, respectively. 

Problems 72 to 76 refer to a river which flows at the rate of 3 miles per hour 
and to a man who rows on the river with such force that he would go 4 miles 
per hour in still water. In each problem draw a preliminary figure showing 
vectors for the velocities of the stream and the man. 

72. The man aims his boat 65° from straight downstream. In what 
direction does he travel and with what speed? 

73. The man aims his boat 75° from straight upstream. In what di¬ 
rection does he actually travel and with what speed? 

74. In order to row straight across the river, in what direction should 
he aim his boat? 

75. In order to row straight at a point whose direction from him is 
inclined 48° from downstream, in what direction should he aim his boat? 

76. In order to row straight at a point whose direction from him is 
inclined 70° from upstream, in what direction should he aim his boat? 


I 


OBLIQUE TRIANGLES 


147 


SUPPLEMENTARY PROBLEMS 

77. The area of a triangle is 1250 square feet and the lengths of two 
sides are 52.8 feet and 63.4 feet. Solve the triangle. 

78. The area of a triangle is 8346 square feet and two of its angles are 
37° 25' and 56° 17'. Solve the triangle. 

79. The building regulations in a city specify that a two-family dwelling 
may not be placed on a lot whose area is less than 15,000 square feet. 
The angle at the street corner of a triangular corner lot is 78° 36' and the 
frontage of the lot on one street is 163 feet. What frontage must be ob¬ 
tained on the other street to permit the erection of a two-family dwelling? 

80. The angles of a triangle are to each other as 3 is to 5 is to 4 and the 
shortest side is 18.5 feet long. Solve the triangle. 

81. The radius of the inscribed circle in a triangle is 6.8 feet and the 
triangle’s area is 496 square feet. Find the sides if one angle is 57° 26'. 

82. The area of a triangle is 839 square feet. One side is 35.73 feet 
long and an adjacent angle is 36° 42'. Solve the triangle. 

83. The sides of a triangle are to each other as 3 is to 5 is to 6 . If the 
area of the triangle is 583 square feet, find the lengths of the sides. 

84. Prove that the area of any parallelogram equals the product of the 
lengths of a pair of adjacent sides times the sine of their included angle. 

85. Prove that the area of any convex quadrilateral is equal to one half 
of the product of the lengths of its diagonals times the sine of either angle 
formed by the diagonals. 

86 . Let R represent the radius of the 
circumscribed circle for a triangle ABC. 

Prove that 

sm a sin p sin y 

Hint. 1 . Let a be any angle and prove that 
2 R = a/sin a. 

2. Through O, the center of the circle, draw 
the diameter COD. 

3. Let ZCDB = 9, and suppose that a is 
acute, as in Figure 91. Then 0 = a because 
9 and a subtend the same arc ( angles at the circumference of a circle are pro¬ 
portional to their subtended arcs). 

4. In right triangle CDB, DC = 2 R. Hence, CB = 2 R sin a. 

5. The student should draw a figure and modify the preceding details for the 
case when a is obtuse. 



87. Prove that, in any triangle ABC, ~ = sin a + sin /3 + sin y. 






148 


TRIGONOMETRY 


88 . By use of half-angle formulas, prove that 


R = 


abc 

4Vs(s^j(^6)(s^c) 


> or 


K = 


abc 
4 W 


where K is the area of triangle ABC. 

89. Prove that the radius of the inscribed circle of any triangle ABC is 
given by the formula r = s tan %a tan 4/3 tan 4 y. 

abc a ft 

90. In any triangle ABC, prove that K = — cos ^ cos ^ cos t, - 

. B . 7 a 

91. In any triangle ABC, prove that r = a sm ^ sm ^ se c 


92. A flagpole of height h feet at the top of a wall inclines at an angle a 
from the vertical toward an observer. From a point A on a level with the 
base of the wall, the angles of elevation of the bottom and the top of the pole 
are observed to be /3 and y, respectively. Derive a formula for the height 
of the wall in terms of h, a, /3, and y. 

93. A statue of height h feet stands on a pedestal whose height is k feet. 

At a certain point A on a level with the base 
of the pedestal, it and the statue subtend the 
same angle, a. (1) Obtain a formula expressing 
the distance, x, from A to the pedestal in terms 
of h, k, and a. (2) Obtain a trigonometric equa¬ 
tion from which a could be found if h and k 
were given. 

Hint. In Figure 92, eventually apply the law 
of sines to the triangle ACD. Write equations 
Fig. 92 involving the letters in the figure and solve for the 

desired quantities. 



94. A boat is sailing due north toward a shore which runs due east 
and west. The point of the shore nearest to the boat is A. On the shore, 
B is 16 miles east and C is 23 miles east of A. At what distance from the 
shore will an observer on the boat find that the line BC subtends an 
angle of 7° 20'? 

95. A flagpole 25 feet high stands on the top of a tower which is 105 feet 
high. At what distance from the base of the tower will the flagpole sub¬ 
tend an angle of 3° 20'? 


96. A flagpole rises from a horizontal plane. In this plane, there are 
two points, A and B, h feet apart and due east of the pole. A and B are 
equidistant from two points C and D, respectively, on the pole. The angle 
of elevation of C from A is a, and of D from B is /3. Derive a formula for 
the distance CD in terms of h, a, and /3. 






★CHAPTER X* 

POLAR COORDINATES 

104. Polar coordinates. In a given plane, the position of any 
point can be described by telling its distance and direction from some 
fixed point. This is the basis for the defi ni tion of polar coordinates. 

In the given plane, let 0 be a fixed point, called the pole, and 
let OX be a fixed half-line, called the polar axis. On OX, in Figurh 93, 
specify a unit for measuring linear dis¬ 
tance. Let P be any point in the plane. 

Let d be any angle whose initial side is 
OX and whose terminal side falls on the 
line through 0 and P. Let r be the dis¬ 
tance OP, considered positive or negative 
according as P is on the terminal side of 6 
or on the extension of this side through 0. 

Then, we call r and 6, together, a set of polar 
coordinates for P. We call r the radius 
vector and d the vectorial angle of P. 

Illustration 1. To plot the point M : (r « 3, 6 = 120°) in Figure 93, draw 
the terminal side of d; measure OM = 3 along this side. N is (— 3, 120°). 

Suppose that 6 is any particular vectorial angle for P. Then, 
other possible vectorial angles are (0 + k • 180°) where k may be any 
integer, because the terminal side of any one of these angles falls 
on the line through OP. Hence, any point P has infinitely many 
different sets of polar coordinates corresponding to a given pole 0 and 
polar axis OX. In contrast, P would have just one set of rectangular 
coordinates with respect to given axes. 

Illustration 2. In Figure 93, M is (r = 3, 0 — 120°). Other coordinates for 
M are (3, 480°), (3, - 240°), (- 3, 300°), etc. 

105. Relation between polar and rectangular coordinates. Let 

OX and OF be a set of rectangular axes. Choose 0 as the pole for 
a system of polar coordinates, and the positive side OX of the z-axis 
as the polar axis. Then, in Figure 94, for any position of P its rec- 

* Not a prerequisite for any later chapter in this book. 

149 




150 


TRIGONOMETRY 



tangular coordinates ( x, y ) and its polar 
coordinates (r, 6) are related by the follow¬ 
ing equations: 

r = ± Vx 2 + y 2 ; tan 0 = (1) 

x = r cos 0; y — r sin 0. ( 2 ) 

Illustration 1. If P is the point (r = 3,6 = 150°), 
then by use of ( 2 ) we find 

x = 3 sin 150° = f; y = 3 cos 150° = - |V§. 


Example 1. Find polar coordinates for the point (x = 3, y = — 4). 


Solution. 1. Since P is in quadrant IV, in Fig¬ 
ure 95, we shall choose 6 as an angle in quadrant 
IV with tan 6 = — f = — 1.333. [Using (l)]] 

2. From Table VII, tan 53° 8' = 1.333. 
Hence, 6 = 360° - 53° 8' = 306° 52'. 

3. From (1), r = V 9 + 16 = 5. The polar 
coordinates are (5, 306° 52'). 

4. Another set of coordinates is 

(r = - 5,9 = 126° 52'). 

Comment. When we are allowed to select the 
polar coordinates of a point P, usually we choose 9 
so that r is positive and 0° g 6 < 360°. 


Y 



EXERCISE 57 


(1) Plot the point whose polar coordinates are given, and give another set 
of coordinates in which r and 9 are positive. (2) Find the rectangular coordi¬ 
nates for the point, with the axes chosen as in Figure 94. 


1 . (3, 180°). 6 . (- 2, 7r). 

2. (2, 360°). 7. (3, U). 

3 . (5,0°). 8. (2,ix). 

4. (5, 90°). 9. (V2, - 45°). 

5. (1, |tt). 10. (2, - 60°). 


11. (- 3, 90°). 

12. (- 1, 30°). 

13. (- V 2 , 135°). 

14. (V 2 , - 135°). 

15. (3, 72°). 


16. (4, 65°). 

17. (-2,230°). 

18. (- 3, 340°). 

19. (6, 105°). 

20. (- 4,205°). 


Each pair of numbers gives the rectangular coordinates of a point. Plot 
the point and find two sets of polar coordinates for it, with r positive in one 
set and negative in the other: 

21. ( 1 , 1 ). 22. (Vs, 1 ). 23. ( 3 , 0 ). 24. ( 2 , 2 ). 25. ( 0 , 2 ). 

26. (- 2, 0). 28. (Vs, - 1). 30. (3, - 3). 

27. (- 1, - 1). 29. (- 1, - Vs). 31. (0, - 2). 

32. ( 1 , Vs)'. 34. ( 3 , 4 ). 36. (- 24, - 7 ). 38. (- 15 , 8 ). 

33. (4, 3). 35. ( 8 , 15). 37. (- 3, - 4). 39. (7, - 24). 











POLAR COORDINATES 


151 


106. Graphs in polar coordinates. Suppose that two variables 
r and 6 represent the polar coordinates of a variable point. Then, 
the graph of an equation relating r and 6 consists of those points 
in the plane whose coordinates (r, 6) satisfy the equation. As a particu¬ 
lar case, the equation may involve only one of the variables (r, 6). 

Illustration 1. The graph of the equation r = 3 is a circle whose radius is 3 
and center is at the pole 0. 


Illustration 2. The graph of the equation 0 = 30° is the whole straight line 
through 0 formed by the terminal side of 30° and its extension through the pole. 

Example 1. Graph r = cos 20. 

Solution. 1. We assign values to 0 and compute the corresponding values of r, 
given in the following table. We plot (r = 1, 0 = 0°), (r = .9, 0 = 15°), etc., 
in Figure 96, and draw the curve 
ABOCD through the points. 


r 

1.0 

.9 

.5 

0.0 

-.5 

-.9 

- 1.0 

e 

0 ° 

15 ° 

30 ° 

45 ° 

60 ° 

75 ° 

90 ° 


2. On assigning values to 0 from 
0 = 90° to 0 = 180°, we obtain the 
curve DEOFH. The remainder of 
the graph is obtained for values of 
0 from 180° to 360°. On account of 
the periodicity of the cosine, if any 
other value is assigned to 0, the 
point (r, 0) obtained will fall some¬ 
where on the graph already obtained 
for values of 0 from 0° to 360°. The 
complete graph is called a four- 
leafed rose. 



D * 

Fig. 96. r = cos 20 


EXERCISE 68 

Graph each equation on polar coordinate paper: 

1. r = 5. 2. r = 3. 3. 0 = 20°. 4. 0 = 150°. 5. r = sin 0. 

6. r = cos 0. 9. r = 1 — sin 0. 12. r = cos 30. 15. r = sin 30. 

7. r = — 2 cos 0. 10. r = 1 + cos 0. 13. r = 5 sec 0. 16. r = tan 0. 

8. r = — 3 sin 0. 11. r = sin 20. 14. r — 3 esc 0. 17. r = cot0. 

2 ._4_6 


18. r = 


2 — cos 


19. r = 


sin 0 


20 . r = 


3 — sin 


21. With 0 expressed in radians, graph the equation r = 30 from 
= 0 to 0 = 2 t r. 


















CHAPTER XI 

INVERSE TRIGONOMETRIC FUNCTIONS 

107. Inverse trigonometric functions. If x = sin y, then y is 
an angle whose sine is x. Corresponding to each value which may 
be assigned to x, we can determine infinitely many values for y. 
Hence, we say that the equation x = sin y defines y as an infinitely 
many valued function * of x. 

Illustration 1. If \ = sin y, then 30° and 150° are the solutions between 0° 
and 360°. Any solution is of the form y = 30° + n ■ 360° or y = 150° + n • 360°, 
where n may be any integer; there are infinitely many solutions. 

Definition I. The expression arcsin x means an angle whose sine 

is x. That is, each of the following equations states the same fact: 

x = sin y and y = arcsin x. (1) 

Illustration 2. If y = arcsin §, then sin y = \. Hence, from Illustration 1, 
arcsin \ = 30°, 150°, 390°, 510°, — 330°, — 210°, etc. It can be verified that a 
general formula for these values is 

arcsin \ = n • 180° + (— l) n (30°), where n is any integer. 

The equation x = sin y describes x as a function of y, and the equa¬ 
tion y = arcsin x performs the inverse duty of describing y as a 
function of x. This leads us to call sin y and arcsin x inverse func¬ 
tions. We call arcsin x an inverse trigonometric function of x, the 
inverse of the sine. 

In arcsin x, the variable x represents the sine of an angle and 
hence — 1 ^ x ^ 1. That is, arcsin x is defined only for values of x 
between — 1 and + 1, inclusive. To each of these values of x there 
correspond infinitely many values of arcsin x. 

Note 1. In place of arcsin x, a common notation is f sin -1 x. For abbrevia¬ 
tion, either arcsin x or sin -1 x may be read the arc-sine of x, or the inverse-sine 
of x, or the anti-sine of x. At present, it is best to read arcsin x or sin -1 x without 
abbreviation as an angle whose sine is x. 

The student should write complete descriptions like Definition I 
for arccos x, arctan x, arccot x, arcsec x, and arccsc x. 

* Recall the general definition of a function in Section 2, page 1. 

t Remember that, in sin -1 x, — 1 is not an exponent. To indicate the minus first 
power of sin x we would write (sin a;) -1 . 


152 


INVERSE TRIGONOMETRIC FUNCTIONS 


153 


Example 1. Find arccos (— ^). 

Solution. 1. Let y = arccos (— J). Then, cos y = — 

2. Since cos 60° = hence y = 120°, 240°, or any angle obtainable by adding 
any integral multiple of 360° to 120°, or to 240°. That is, 

arccos (- £) = 120°, 240°, - 120°, - 240°, 480°, 600°, etc. (2) 

[A formula for (2) is arccos (— §) = 2n7r ± |7T, where n is any integer.] 


Example 2. Find tan arccos tz if arccos tw is in quadrant IV. 

Solution. 1. Let arccos ts = 9; then cos 9 = ^ and 9 is in quadrant IV. 


2. tan 9 = 


sin 9 
cos 9 


- ^1 - cos 2 9 Vl - J 


cos 9 


25 12 

= —g- • That is, 


tan arccos ^ if arccos ^ is in quadrant IV. 


Example 3. Find sec arctan x. 

Solution. Let 9 = arctan x; then tan 9 = x. Hence, 

sec arctan x = sec 9 = ± ^1 + tan 2 9; or, sec arctan x = ± Vl x 2 - 

Note 2. In general, any equation involving two variables x and y defines 
either variable as a function of the other. The two functions thus defined are 
called inverse functions; either is called the inverse of the other. This notion 
includes inverse trigonometric functions as a special case. Another important 
case involves the equation y = a x . If a > 0, then x = log a y. Hence, the ex¬ 
ponential function, a x , is the inverse of the logarithm function, logo y. 


EXERCISE 59 

Write equivalent equations in the notation of inverse functions: 

1. tan y = f. 3. esc y — — 5. 5. sin d = }. 7. cot 6 = x. 

2. sin y = 4. tan y — 6. sec y = — 4. 8. sec 9 = w. 


10 . 

17. 

18. 

19. 

20 . 
21 . 
32. 


12 . 


Find * two positive or zero values and two negative values for each expression, 
if possible; stale the results in radian measure: 

9. arccos 1. 11. arctan 1. 13. arccos 0. 

arcsin 0. 14. arccot 1. 

22. arctan V3. 27. 

23. arccot 0. 28. 

24. arcsec V 2. 29. 

25. arccot (— V^). 30. 

26. arccot (— iV^l). 31. 


arcsin 1. 
arcsin 

arccos (— |). 
arctan 0. 
arcsin (— ^v^). 
arccos (|V3). 
arcsin 3. 33. 


arccos 2. 


34. arcsec 


* The instructor may ask for general formulas for all values of 


15. arcsec 2. 

16. arccsc 1. 
arccos (— 
arcsin (— fvll). 
arccsc V2. 
arccsc (— V2). 
arcsin 

35. arccsc i. 
the expressions. 









154 


TRIGONOMETRY 


Find two positive values for the expression, in degree measure: 

36. arcsin .2051. 38. arctan .2247. 40. arccos ( .4924). 

37. arccos .9696. 39. «arccot 1.560. 41. arcsin (— .8371). 

Assume that any angle involved is between 0 and 180 , and find the specified 


function: 






42. 

sin arcsin f. 

50. 

tan arcsin f. 


58. 

esc arcsin x. 

43. 

cos arccos |. 

51. 

cot arccos 


59. 

sec arccos x. 

44. 

tan arccot 

52. 

sec arctan ( — 

t). 

60. 

tan arcsin x. 

45. 

cot arctan 5. 

53. 

esc arccot -ft-. 


61. 

cot arccos x. 

46. 

sec arccos 

54. 

esc arccot (— 

4). 

62. 

sin arctan x. 

47. 

esc arcsin f. 

55. 

sec arctan 


63. 

cos arccot x. 

48. 

sin arccos f. 

56. 

cos arcsin z. 


64. 

sec arctan x. 

49. 

cos arcsin 

57. 

tan arccot x. 


65. 

esc arccot x. 

66. Prove that (1) 

arccos 

x is defined only for values 

of x between — 1 


an y _|_ i ; inclusive; (2) arcsec x and arccsc x are defined only for values of 
x which are numerically greater than or equal to 1; (3) arctan x and arccot x 
are defined for all values of x. 

Carry the solution of each equation only to a stage where all values of the 
unknown can be stated by use of inverse trigonometric functions: 

67. 6 sin 2 x - 7 sin x - 3 = 0. 70. 6 sin x - esc x + 1 = 0. 

68. 12 cos 2 x + 5 cos x - 2 = 0. 71. 8 cos x - 3 sec x + 10 = 0. 

69. 2 tan x + 15 cot x = 13. 72. 4 cot x + 15 tan a? - 23 = 0. 

108. Principal values. We recall that, if y = arcsin x, then in¬ 
finitely many values of y correspond to any given value of x. To 
avoid confusion, it will prove convenient to designate a certain one 
of these values of y as the principal value of arcsin x. Similar re¬ 
marks apply to the other inverse trigonometric functions. 

★Note 1. The main justification for the notions of the present section is met 
in the more advanced course called Integral Calculus where certain fundamental 
integrals are expressed in terms of principal values of inverse trigonometric func¬ 
tions. Hence, the following definitions are arranged * to suit the needs of calculus. 

The succeeding definitions will be found to imply that 

if x ^ 0, the principal value of any inverse trigonometric function | ^ 

of x is that single value lying between 0 and ir/2, inclusive. / 

* This prevents a single simple definition for all principal values and, in particular, 
causes apparent artificiality in the definitions relating to arcsec x and arccsc x. See 
pages 105-109, Granville’s Calculus (1929). Ginn and Company, publishers. 


INVERSE TRIGONOMETRIC FUNCTIONS 


155 


Illustration 1. The principal value of arcsin \ is 30°, or 7r/6. 

For any given value of x, the following definitions apply. 

Definition II. The 'principal value of arcsin x or arctan x is that 
value which is smallest numerically (this value is negative and not 
less than — tv/ 2 if x < 0). 

Definition III. The principal value of arccos x or arccot x is the 
smallest positive value of the function (this value is between 0 and t, 
inclusive). 

Example 1. Find the principal value of arcsin (— \). 

Solution. Since sin 30° = §, hence arcsin (— J) = 210°, 330°, — 30°, etc. 
By Definition II, the principal value is — 30°, or — 7 t/6 . 

In writing, we shall commence the name of an inverse function 
with a small letter to indicate any value of the function, and with a 

capital letter to indicate the principal value. 

Illustration 2. From Example 1, Arcsin (— ■§) = — \ir. We read this “the 
arcsin of — J is — 7t/6.” 

Example 2. Find Arccos (— |). 

Solution. Since cos 60° = §, hence arccos (— f) = 120°, 240°, — 120°, etc. 
By Definition III, Arccos (— J) = 120°, or 2tt/3. 

The following table summarizes Definitions II and III and also 
defines Arcsec x and Arccsc x. 


Principal Values of Inverse Functions 


x 0 

0 ^ ( any principal value ) 5S 

t-H 

VII 

H 

VII 

t-H 

1 

— ^7T ^ Arcsin x ^ r 

t-H 

VII 

H 

VII 

t-H 

1 

0 ^ Arccos x ^ 7r 

Any value of x 

— -|7r < Arctan x < r 

Any value of x 

0 < Arccot x < i r 

x ^ 1 

0 ^ Arcsec x < 

x — 1 

— 7 r ^ Arcsec x < — 

x ^ 1 

0 < Arccsc x ^ 7t/2 

x ^ — 1 

— 7r < Arccsc x -A — 


Note 2. Any inverse trigonometric function of x is an infinitely many valued 
function of x, whereas the principal value of the function has a single value for 
each value of x. This single-valuedness of Arcsin x, Arccos x, etc., largely accounts 
for their usefulness as compared to the functions arcsin x, arccos x, etc. 












156 


TRIGONOMETRY 


EXERCISE 60 

State each principal value both in degrees and radians: 

1. Arcsin 4. Arctan 1. 7. Arccot Vlj. 10. Arccot 0. 

2. Arccos 1. 5. Arcsec V2. 8. Arccos QV 3 ). 11. Arcsin 0. 

3. Arcsin 1. 6. Arccsc 2. 9. Arctan 0. 12. Arccos 0. 


13. 

Arcsin (—1). 

18. 

Arcsin (— 

23. 

Arctan (^V 3 ). 

14. 

Arccos (— ^v^ 2 ). 

19. 

Arccos (—1). 

24. 

Arccsc (—1). 

15. 

Arctan (—1). 

20. 

Arcsin (|V2). 

25. 

Arcsec (—2). 

16. 

Arccot (—1). 

21. 

Arctan (— 3 V 3 ). 

26. 

Arccsc (—2). 

17. 

Arccot (— V 3 ). 

22 . 

Arccot (— 1V^3). 

27. 

Arcsec (—1). 

Find the principal value, by 

r use of Table YII: 



28. 

Arcsin .0814. 

30. 

Arccot .1793. 

32. 

Arcsin (— .2486). 

29. 

Arccos .9652. 

31. 

Arccot 3.108. 

33. 

Arctan (— .3068) 

Find the value of each quantity without tables: 



34. 

sin Arcsin 1. 

37. 

sin Arccos 

40. 

cos Arcsin 

35. 

cos Arccos |. 

38. 

sec Arccos f. 

41. 

sin Arccos (— 5 ). 

36. 

cot Arccos 5 . 

39. 

cot Arcsin f. 

42. 

cot Arccos (— |). 

43. 

cos Arcsin (— §). 


62. esc Arctan (— v'jj). 

44. 

tan Arctan (—3). 


53. cot Arccos (—-&). 

45. 

cot Arccot (— 6 ). 


54. sin Arctan ( 

:-¥). 

46. 

sin Arctan (—1). 


55. cos ( 7 r 

— Arccos i). 

47. 

cos Arccot (—1). 


56. cot[V 

+ Arctan ( — 5)3. 

48. 

sec Arccot (v 3 ). 


57. cos (90 

0 + Arcsin |). 

49. 

cos Arcsin (—1). 


58. tan (9C 

1° - 

Arcsin xf) . 

50. 

tan Arcsin (— 5 V 2 ). 

59. sin [270° + Arccos (- f)]. 

51. 

sin Arccot (J^3). 


60. cos [270° - 

Arcsin (— f)]. 

61. 

sin (2 Arcsin f). 

62. 

cos (2 Arcsin i). 

63. 

tan (2 Arctan 3). 

Hint for Problem 61. 

Let Arcsin f = 6. We desire sin 

29. 

64. 

sin (2 Arcsin -fr ) . 

68. 

cos (i Arcsin xf) • 

72. 

Arcsin (sin 140°). 

65. 

cos (2 Arccos A). 

69. 

tan (§ Arccos |) . 

73. 

Arccos (cos 250°). 

66. 

cot (2 Arctan 2). 

70. 

Arcsin (sin 60°). 

74. 

Arctan (tan 105°), 

67. 

sin (^ Arccos f). 

71. 

Arccos (cos 68 °). 

75. 

Arcsin (sin 200°). 


76. sin (Arcsin x — Arcsin y ). 

77. cos (Arccos x + Arcsin y). 


78. tan (Arctan x + Arctan y ). 

79. tan (Arccot x — Arccot y). 


INVERSE TRIGONOMETRIC FUNCTIONS 


157 


if Find the specified quantity , if x, y, and z are positive: 

(Arccos 3 y + Arccos 2x). 
(Arctan x + Arccot 2y). 


80. cos (Arccos x — Arccos 3 y). 82. sm 

81. cot (Arccot x — Arctan y). 83. cot 

84. sin (2 Arcsin x). 87. cot (2 Arctan x). 90. tan 2 (| Arccos y). 

85. cos (2 Arccos x). 88. sin 2 (£ Arccos x). 91. sin (^ Arcsin x). 

86. tan (2 Arctan x). 89. cos 2 (| Arccos x). 92. cos (^ Arcsin x). 

93. sin (Arcsin x + Arccos y + Arcsin z). 95. sin (3 Arcsin x). 

94. cos (Arcsin x + Arcsin y + Arccos z). 96. cos (3 Arccos x). 

'kFind the inverse function if a and j3 are between 0° and 45°: 

97. Arccos (1—2 sin 2 a). 99. Arcsin (sin a cos (3 — cos a sin j8). 


98. Arcsin 


ft 

— cos a 

J 

2 


100. Arcsin 


in -\j- 


+ cos a 


★109. Graphs of the inverse trigonometric functions. Since the 
equation y = arcsin x is equivalent to x = sin y, the graph of the equa¬ 
tion y = arcsin x in Figure 97 is a sine curve along the y-axis. 

Note 1. Look at Figure 97 through the other side of this sheet, held against 
a light, to orient the curve as in Figure 59, page] 82. A set of points for obtain¬ 
ing Figure 97 can be found by interchanging x and y in the table for Figure 59. 

The graph of y = arccot x in Figure 98 is identical with the graph 
of x = cot y and hence is related to Figure 63 on page 85. 

In Figure 97 the graph of the principal value, y = Arcsin x, is 
simply the arc AOB. Similarly, the graph of y = Arccot x is simply 
the branch ABC in Figure 98. 



Y 



Fig. 97. y = arcsin x 


Fia. 98. y — arccot x 




















158 


TRIGONOMETRY 


★EXERCISE 61 

Graph the function with y measured in radians. Use the same scale for 
x and y, with the y-axis vertical. Mark the principal value. 

1 . y = arccos x. 2. y = arctan x. 3 . y = arcsec x. 4. y = arccsc x. 

★110. Identities and equations involving inverse functions. 

To solve an equation involving inverse trigonometric functions, 

1 . try to obtain a related algebraic equation by equating the same trig¬ 
onometric function of both sides of the given equation; 

2 . substitute in the given equation all values of the unknowns obtained 
from the algebraic equation, to determine which values are roots. 

Example 1. Solve for x: Arcsin 2x + Arcsin x = 30°. 

Solution. 1. Arcsin 2x = 30° — Arcsin x. 

2. Take the sine of both sides: 2x = sin (30° - Arcsin x); 

2x = jVl - x 2 - f xV3; or, x(4 + v'S) = Vl - x 2 . 

3. Square both sides and simplify: 

x 2 (20 + 8V§) = 1; x = ± .1719. (Using logarithms) 

Check. 1. Substitute x = .1719 in the given equation. From Table VII, 
Arcsin 2x + Arcsin x = 20° 6' + 9° 54' = 30° O'. Hence, .1719 is a solution. 

2. On substituting x = — .1719, we find that it is not a solution: 
if x = — .1719, Arcsin 2x + Arcsin x = — 20° 6' — 9° 54' = — 30°. 

Note 1. An equality involving general values of the inverse trigonometric 
functions may be true in the sense that 

any value of either side equals some value of the other side; (1) 

or, any value of just one side equals some value of the other side. (2) 

The equality may be an identity for principal values of the functions without either 
(1) or (2) being true. Or, (1) or (2) may be true without the equality being an iden¬ 
tity for principal values. 

oc 

Example 2. Prove the identity: Arcsin x = Arctan -^===. 

Solution. Find the tangent of Arcsin x: tan Arcsin x = —===. 

Vl - x 2 

Hence, the stated identity is true because its sides represent two angles, each 
between — 90° and 90°, whose tangents are equal. 

Comment. The related equality involving general values of the functions, 

X 

arcsin x = arctan —» (3) 

Vl - x* 

is not true for all values of x, in either of senses 1 and 2 of Note 1. This can be 
verified by writing down sample values of each side of (3) when x = | V2. 










INVERSE TRIGONOMETRIC FUNCTIONS 


159 


Example 3. Prove that any value of the left side equals some value of 

2x 


the right side: 


2 arctan x = arctan 


1 - x 2 


(4) 


Solution. Take the tangent of the left side: tan (2 arctan x) = 


2x 


1 - x 2 

Hence, 2 arctan x is an angle whose tangent is 2x/(l — x 2 ). 

Comment. The reader may verify that, if x = — V3, then it is not true that 
any value of the right side equals a value of the left side. Hence, (1) in Note 1 
is not true for (4). 

★EXERCISE 62 

Prove the identity. If general values of inverse functions enter, show that 
any value of the left side equals some value of the right side. 

x 


1. Arccos x = Arccot 

2. Arccot x = Arccos 


x 

Vl - x 2 

x 


VT+ 


3. Arctan x — Arcsin 


4. Arcsec x = Arccsc 


x 2 


Vl + 


Vz 2 - 1 


15. Arctan x + Arccos 


yVl - 

X 2 

, if x ^ 

0 . 

11. 

i- 

12. 

h 

. 13. 

2 i 

I. 14. 

3£ 

X 



7. 2 arccos a: = arccos (2x 2 — 1). 

;0. 12. \ Arccos x = Arcsin 

x 2 - 1 
2x 

10. Arcsec x +Arccsc x = 90°, if x^O. 14. 3 arcsin x = arcsin (3x — 4x 3 ). 

7T 


- = -> if x ^ 0. 

Vl + X 2 2 


Solve for x and check by substitution: 

16. Arccot x = Arctan x. 

17. Arcsin x = Arccos x. 

18. Arcsin (2x 2 + x) = \tt. 

19. Arccos (1 — x) = Arccos 2x 2 . 

20. Arctan x — Arccot 3x. 

21. Arctan 3x = 90° — Arctan x. 

22. Arcsin x = Arcsin f + 30°. 

23. Arccos 2x = Arcsin tV + 45°. 

24. Arctan 3x + Arctan x = Arccot \. 

25. Arccot 2x + Arccot * = Arctan 3. 33. Arctan 3x + Arccot x = It. 


26. Arcsin x + Arccos 2x = 30°. 

27. Arccos x — Arccos 2x = 90°. 

28. Arcsin 3x — Arcsin x = 90°. 

29. Arcsin x + Arcsin 2x = 60°. 

30. Arccos x + Arccos 3x = iir. 

31. Arcsin - + Arcsin - = 

x x 2 

32. Arcsin - + Arccos — = 

x x 2 



















★CHAPTER XII 

COMPLEX NUMBERS AND DE MOIVRE’S THEOREM 


111. Com plex numbers. In th is ch apter we shall always use * i 
to denote V— 1. That is, i = V— 1 and by definition i 2 = — 1. 
If a and b are real, then (a + hi ) is called a complex number, whose 
real part is a and imaginary part is bi; we call b the coefficient of the 
imaginary part. Any real number a is considered as a complex num¬ 
ber in which the coefficient of the imaginary part is zero. A com¬ 
plex number (a + bi) is called an imaginary number if b 0. If a = 0 
and b 0 , then (a + bi) is called a pure imaginary number. 

Illustration 1. In the form of complex numbers, a = a + Of; 6 = 6 + Of; 
0 = 0 + Of. The expression (3 -f 5 i) is an imaginary number; 6f is a pure 
imaginary number. 

Definition I. Two complex numbers ( a + bi) and (c + di) will be 
called equal, in case a = c and b = d. 

Since 0 = 0 + 0 i, it follows from Definition I that 

if (a + bi) = 0, then a = 0 and 6 = 0. (1) 

We agree that, f in any indicated sum, difference, product, or quo¬ 
tient involving integral powers of complex numbers, the result shall 
be that which is obtained by operating as if i were a literal num¬ 
ber subject to the rules of algebra for real numbers. In particular, 
any integral power of i can be reduced to either i, — i, 1 , or — 1 . 

Illustration 2. f 3 = f(f*) = f(- 1) = - f; f 4 = (f 2 ) 2 = (- l) 2 = 1. 

(3 + i) (5 + 2 i) = 15 + Ilf + 2f 2 = 15 + Ilf - 2 = 13 + Ilf. 

If P is any positive number, the square roots of (— P) are iVp 
and — isfp because 

(iVp) 2 = i 2 P = - P; (- iVp) 2 = i 2 P = - P. 
Hereafter, the symbol V— P, whichjs read the square root of — P, 
will represent the particular root iVP; that is, V — P = ix^P. 

Illustration 3. V— 4 = f+4 = 2f. The square roots of — 9 are ± 3f. 

* Except where otherwise specified, all other literal numbers will represent real 
numbers. 

t For a logical foundation of the algebra of complex numbers, see Dickson’s Elemen¬ 
tary Theory of Equations, page 1. 


160 




COMPLEX NUMBERS 


161 


112. Conjugate complex numbers. If two complex numbers 
differ only in the signs of their imaginary parts, the two numbers 
are called conjugate complex numbers, and either is called the 
conjugate of the other. 

Illustration 1. The conjugate of (a — bi ) is (a + hi), and 

(o — bi)(a + bi) = o 2 — (bi) 1 = a 2 — bH 2 = a 2 + b 2 ; 

(a + bi) + (a — bi) = 2 a; (a + bi) — (a — bi) = 2 bi. 


From Illustration 1 it is seen that the sum and the product of two 
conjugate complex numbers is real, and their difference is a pure 
imaginary number. 

A quotient of two complex numbers can be reduced to the form 

(a + bi) by multiplying both numerator and denominator by the con¬ 
jugate of the denominator. 


Example 1. Reduce to the form (a + bi): 


5 + 2 i 
3 - 4i‘ 


Solution. 

Hence, 


5 + 2 i 5 + 2t 3 + 4i 15 + 26i + 8 i 2 

3 — 4i “ 3 — 4 + 3 + 4£ 9 — 16i 2 

5 + 2 i 15 + 26i - 8 _ 7 , 26 . 

3 — M ~ 9 + 16 25 + 25 


EXERCISE 63 

Simplify each power of i to either i, — i, 1, or — 1: 

1 2 i 7 3. i 10 4- f 13 . 5* i 9 - 6. i s . 

Tell the two square roots of each number: 

7. _ 49. 8. - 144. 9. - 25. 10. - 32. 11. - 50. 

Perform indicated operations and simplify to the form (a + bi ): 


12. 

(2 + Si) - 

(5 - V^4). 

13. (4 + 

V^9) 

- 

(3 

V 

- 25). 

14. 

Si(5i 2 ). 

15. 2i(7i 3 ). 

16. \ 

^sV- 

27. 

17. 


^2' 

v++8. 

18. 

(3 + 5i)(6 

-7 0. 

21. 

(V+^2 

+ 3)(v 

r ~~ 

2 - 

-3). 


19. 

(2 - 40(3 

+ tt). 

22. 

(4- V' 

Zr 5)(4 

+ 


- 5). 


20. 

(2 - 40(2 

+ 4f). 

23. 

(V^3 

+ 0(3 

- 

V- 

- 12), 


24. 

2 + 3 i 

5 + 4 i 

"■ m 

28. 

hi 

1 + 7 i 

30. 

3 

2 % 


32. 

5 

i 7 

25. 

3 - hi 

4 + Si 

»• !^r 

29. 

6 

1 + Si 

31. 

1 

i 3 


33. 

6 

¥ 


34. Prove that, if a 2 + b 2 = 1, the reciprocal of (a + bi) is (a - bi). 


















162 


TRIGONOMETRY 


Find x and y by use of Definition I and statement 1, page 160: 

35. x + yi = 3 + 2 i. 37. x + 2 -\- yi — Si = 0. 

36. 2x + yi = 5 — 3 i. 38. 2x — bi — 4 + yi = 0. 

Express the reciprocal of the number in the form (a + bi ): 

39. 3 + 5 i. 40. 5 + V^5. 41. Si. 42. V^~5 + V- 20. 


113. Geometrical representation of complex numbers. On a co¬ 
ordinate system, as in Figure 99, we may represent any complex 
number (a + bi) by the point P whose coordinates are (x = a, y = b). 
The point (a, b) falls on OX when and only when b = 0 and hence 
(a + bi) is a real number; therefore we call OX the axis of real 
numbers. Similarly, ( a + bi) falls on OY when and only when 
a - 0 and hence (a + bi) is a pure imag¬ 
inary number. Therefore we call OY 
the axis of pure imaginary numbers. 

Illustration 1. In Figure 99, A represents 
(— 3 + 6i); B represents — 5, thought of as 
(— 5 + 0 ■ i); C represents — Si. 

It is sometimes useful to think of the 
vector OP, instead of merely the point P , 
as representing ( a + bi). 


(—3+6i)T- 



114. Geometrical addition of two complex numbers. Let M and 

N, in Figure 100, represent (a + bi) and (c + di), respectively. Com¬ 
plete the parallelogram with OM and ON as adjacent sides. Then , 
the fourth vertex, P, represents £(a +‘c) + (b + d)f}. 


Proof. For any location of M and N, if the 
corresponding figure is labeled like Figure 100, 
then 

OR = OS + SR = a + c; 

RP = SM + KN = b + d. 

Hence, P represents [(a + c) + (6 + d)i]. 

Note 1. Let us think of each complex number 
in Figure 100 as a vector. Then, we see that the 
vector representing the sum of two complex num¬ 
bers is obtained by adding the vectors for these numbers 
in accordance with the parallelogram law for vector 
addition. 


Y 



To subtract (c + di) from (a + bi), geometrically, we geometri¬ 
cally add (— c — di) to (a + bi). 











COMPLEX NUMBERS 


163 


EXERCISE 64 


Construct each sum or difference graphically; check algebraically: 


1. (3 - t) + (2 + 3 i). 

2. (2 + 3i) + (1 - 2 i). 

3. (- 3 + 4 i) + (2 i - 5). 

4. (- 6 + 2 i) + (- 7% - 4). 

5. (3) + (6» - 7). 

6. (5i) + (3 - 2 i). 


7. (7 + 0 i) + (0 + 8 i). 

8. (- 12i) + (- 6). 

9. (3 + 8 i) - (3 - Si). 

10. (4 - Si) - (3i - 6). 

11. (3i) - (Si + 4). 

12. (4 + 2i) - (6). 


13. Add entirely geometrically: (3 — 3i) + (2 + 6i) + (— 3 + 4i). 

14. Add entirely geometrically: (— 5 + 2i) + (4 — 3i) + (2 — i). 


Y 



115. Trigonometric form of complex num¬ 
bers. In Figure 101, P represents (a + hi) and 
the units of length on the axes are the same. Let 
r = OP and let d be any angle whose initial side 
is OX and terminal side is OP. Then, for any 
values of a and b, 

r = Va 2 + b 2 ; tan 0 = (1) 

a = r cos 0; b = r sin 0; (2) 

a + bi = r(cos 0 + i sin 0). (3) 


We call r(cos 6 + i sin 6) the trigonometric, or polar, form of 
(a + bi). The positive radius vector r is called the absolute value, 
or modulus, of (a + bi), and 6 is called its amplitude, or argument. 
If 6 is one amplitude, then the other permissible amplitudes are 
(6 + k • 360°) where k may have any integral value, because the 
angle (6 + k • 360°) has the same terminal side as 6. In writing a 
number in polar form, we usually choose the smallest permissible 
positive or zero amplitude. 

Illustration 1. If 20° is one amplitude, then others are 380° and — 340°. Thus, 
5(cos 20°+i sin 20°)=5(cos 380°+f sin 380°) = 5[cos ( — 340°)+f sin ( — 340°)]. 


If two complex numbers are equal, then their moduli are equal and 

their amplitudes are either equal or else differ by some integral 
multiple of 360°, because the given numbers would be represented 
by the same point in Figure 101. 

Illustration 2. To change from the polar form to the form (a + bi), we use 
trigonometric tables. Thus, by use of Table IV, 

6(cos 35° + i sin 35°) = 6(.819 + .574i) = 4.914 + 3.444L 






164 


TRIGONOMETRY 


To express a given number ( a + bi) in its trigonometric form, 
first plot the number and notice what quadrant 6 is in; then find r and 
6 by use of equations 1. 

Example 1. Find the trigonometric form of (5 — 6i). 

Solution. r = Vei, and 9 is in quadrant IV. 

Tan 6 = — | = — 1.200; from Table VII, 9 = 309° 49'. 

Hence, (5 — 6 i) — v / 61 (cos 309° 49 r + i sin 309° 49 / ). 

Note 1. We may write 0 = 0(cos 9 + i sin 9), where 9 has any value. Thus, 
0 may have any angle 9 as its amplitude and the modulus of 0 is 0. 

Note 2. The student who has studied polar coordinates will recognize that 
r and 9 in Figure 101 are a set of polar coordinates for the point P with OX as the 
polar axis. This accounts for calling r(cos 9 + i sin 9) the polar form. 

To plot r(cos 6 + i sin 0), draw a circle with the origin as center and 
r as radius; draw Z.XOS = 6. The point P where OS cuts the circle 
represents r(cos 6 + i sin 6). 

Note 3. In stating a result which is a complex number in polar form, always 
choose the amplitude as a positive or zero angle less than 360°. 


EXERCISE 65 


Plot each number and also express it 


1. 5(cos 60° + i sin 60°). 6. 

2. 3(cos 210° + i sin 210°). 7. 

3. 4(cos 360° + i sin 360°). 8. 

4. 5(cos 180° + i sin 180°). 9. 

5. 6(cos 0° T i sin 0°). 10. 


<i the form (a + bi ): 

7(cos 270° + i sin 270°). 

2[cos (— 45°) + i sin (— 45°)]. 
3[cos (- 150°) + i sin (- 150°)]. 
cos 147° + i sin 147°. 
cos 249° + i sin 249°. 


Reduce each number to its trigonometric form: 

11. 6 + 6 i. 12. hi - 5. 13. »V3 - 1. 14. V 3 - i. 

15. 0 + 3 i. 16. 6 + 0 i. 17. - 71. 18. - 8 . 19. - 10. 

20. - 4 - Si. 22. - 6 + V- 49. 24. cos 50° - i sin 50°. 

21. 3 - 4 i. 23. 4 - V- 25. 25. cos 125° - i sin 125°. 


Change the number and its conjugate to the polar form: 

26. 1 + i. 27. - 1 - i. 28. 4i - 3. 29. 10* - 5. 

30. Tell the amplitude of any positive number; of any negative number; 
of bi if b < 0; of bi if b > 0. 

31. Prove that the conjugate of r(cos 9 + i sin 9) is 

r[cos (— 9 ) + i sin (— 0)]. 




COMPLEX NUMBERS 


165 

116. Theorem. The product of two or more complex numbers, 

in polar form, is a number whose modulus is the product of the moduli 
and whose amplitude is the sum of the amplitudes of the given numbers. 

Proof. 1. If M = r(cos a + i sin a) and N = s(cos /3 + i sin j8), then 
MN = 7-s(cos a cos jS + t cos a sin 0 + i sin a cos 0 - sin a sin (J) 

= rs[(cos a cos 0 - sin a sin 0) + t(sin a cos 0 + cos a sin 0)], or 
r(cos a + i sin a) • s(cos jS + f sin jS) = rs[cos (a + /S) + f sin (a + 0)]. (1) 

This proves the theorem for a product of just two factors. 

2 - If P m t (cos 7 4* t sin 7), then, by use of (1), 

MNP — (MN) • P = rs|[cos (a + / 3 ) + i sin(o; + / 3 )J • i(cos 7+ £ sin 7). (2) 

By use of Step 1 , which applies to any two factors, ( 2 ) becomes 

MNP = rst[_ cos (a + 0 + 7) + i sin (a + 0 + 7)], (3) 

which proves the theorem for' a product of three factors. Similarly, by 
successive applications of Step 1, it follows that the theorem is true for a 
product of any number of factors. 

Illustration 1. By use of the theorem, we find that 

3(cos 40° + i sin 40°) • 5(cos 170° + i sin 170°) = 15(cos 210° + i sin 210°). 

117. De Moivre’s Theorem. If n is any positive integer, 

[r(cos a'+ i sin a)]" = r"(cos na + i sin na). (1) 

Illustration 1. [3(cos 20° + i sin 20°)] 6 = 3 6 (cos 120° + i sin 120°). 

Proof. [r(cos a + i sin a)]” = 

r(cos a + i sin a) • r(cos a + i sin a) . to n factors. (2) 

The product of the moduli in (2) is r • r • • • r, to n factors, or r n ; the sum 
of the amplitudes is (a -f a + • • • + a), to n terms, or na. Hence, by 
Section 116 , equation 1 is true. 

Example 1. Find (1 — £) 4 by use of De Moivre’s theorem. 

Solution. 1. Express (1 — i) in polar form: 
r = V5. tan 9 = — 1 and 6 is in quadrant IV; hence, 9 = 315°. 
Therefore, 1 — i = V2(cos 315° + i sin 315°). 

2. By De Moivre’s theorem, (1 — £) 4 = 

(a/ 2) 4 [cos (4 • 315°) + i sin (4 • 315°)] = 4(cos 1260° + i sin 1260°); 

(1 - iy = 4(cos 180° + i sin 180°) = - 4. (1260° = 3 • 360° + 180°) 



TRIGONOMETRY 


166 

118. Theorem concerning nth roots. If n is any positive integer, 
then any complex number, which is not zero, has exactly n distinct 
nth roots. 

Proof. 1. Let R (cos a + i sin a) be any complex number (with 
R > 0). Let r(cos 0 + i sin 0) be any nth root of the number. Then, 

R (cos a + i sin a) = [r(cos 0 + i sin 0)] n , 
or, by De Moivre’s theorem, 

R( cos a + i sin a) = r n (cos nd + i sin n0). (1) 

2. If two complex numbers are equal, their moduli are equal and their 
amplitudes differ only by some integral multiple of 360°. Hence, r and 0 
satisfy (1) if and only if 

r n = R, and nO = a + k ■ 360°. ( k is any integer) 

Therefore, r = v / R; 0 = ~ + h ■ —— • (2) 

3. On placing k = 0, 1, 2, • • ■, (n - 1) 'in (2), we obtain the following 
n distinct values for 0, all less than ^360° + — ^: 



Corresponding to these amplitudes, with r = V R, we obtain the following 
n distinct nth roots: 

VH(cos l + i sin t)', + ^-) + i (^ + ^)]i etc - 


4. If k is given any value besides 0,1, • • •, (n — 1) in equations 2, the value 
of 0 obtained differs from some one of the angles in (3) only by an integral 
multiple of 360°, and hence gives no additional nth root. Therefore, the 
n roots obtained in Step 3 are the only nth roots of R (cos a + i sin a). 


Summary. To obtain the nth roots of R (cos a + i sin a), place 
k = 0, 1, 2, • • •, n — 1, in the following formula: 



Note 1. All nth roots of f2(cos a + i sin a) have the same modulus, v'k, 
and hence all of these nth roots will be located geometrically on the circum¬ 
ference of a circle whose center is the origin and whose radius is ^R- The 
points representing the roots divide the circumference into n equal parts, because 

360° 

adjacent amplitudes in (3) differ by —• A special case of these remarks is illus¬ 
trated in Figure 102, page 167, in connection with Example 1. 


COMPLEX NUMBERS 


167 


Example 1. Find the cube roots of 8(cos 150° + 

Solution. We substituted = 8, a = 150°, and n = 3 
in (4), and then place k = 0, 1, and 2. The roots ob¬ 
tained are 

2 (cos 50° + i sin 50°); 2 (cos 170° + i sin 170°); 

2 (cos 290° + i sin 290°). 

Comment. The cube roots are represented by P, Q, 
and S in Figure 102. P, Q, and S divide the circum¬ 
ference into three equal parts because the amplitudes 
of P, Q, and S, in succession, differ by 120°. 


sin 150°). 

Y 



EXERCISE 66 

Leave results in polar form unless otherwise directed, or unless the sine 
and cosine involved are known without using tables. 


Find the indicated power by use of De Moivre’s theorem: 


1. [2(cos 120° + i sin 120°)] 4 . 

2. [3(cos 315° + i sin 315°)] 3 . 

5. (3 - 3 i)K 7. (1 + iV 3) 6 . 

6 . (2 + 2i) 4 . 8. (V3 - i) 6 . 


3. [.l(cos 75° + i sin 75 0 )] 6 . 


4. [|(cos 40° + i sin 40°)] 6 . 


9. (3 + 4 i) 6 . 


10. (4 - 3 i) 5 . 


11. (- 7 - 24i) 5 . 

12. (15 - 8 i) 3 . 


Find the specified roots and plot them: 

13. 5th roots of 32(cos 100° + i sin 100°). 

14. 4th roots of 16(cos 240° + i sin 240°). 

15. Cube roots of 1000(cos 300° + i sin 300°). 

16. 5th roots of 32(cos 200° -f- i sin 200°). 

17. 4th roots of 81 (cos 80° + i sin 80°). 

18. Cube roots of 125(cos 210° + i sin 210°). 

19. Cube roots of 1 . 20. 6th roots of 1 . 21. Cube roots of 

22. Cube roots of ( 4 V 2 — 4fv / 2). 27. Square roots of i. 

23. 4th roots of (8 — 8fV3). 28. 

24. Square roots of (— i). 29. 

25. Cube roots of — 8. 30. 

26. Square roots of — 16L 31. 


1 . 


5th roots of (— 32). 

Cube roots of i. 

4th roots of 81. 

4th roots of 8(^2 — W 2). 


Find all the roots of each equation, reduce the results to the form (a + bi), 
and plot them: 

32. x 4 = 16. 33. x 6 + 64 = 0. 34. x 6 = - 243. 35. x 9 - 1 = 0. 

Hint. The roots of x 4 = 16 are, by definition, the fourth roots of 16. 




168 


TRIGONOMETRY 


119. Division of complex numbers in polar form. The modulus 
of the quotient of two complex numbers is the quotient of their moduli, 
and the amplitude of the quotient is the amplitude of the dividend minus 
the amplitude of the divisor. 

Proof. If s ^ 0, then 

r(cos a + i sin a) r (cos a + i sin a) (cos 0 — i sin 0) 


s(cos 0 + i sin 0) s (cos / 3 + i sin 0)(cos /3 — i sin 0) 

_ r (cos a + i sin a;)[cos (— 0) + r sin (— 0)3 
— s cos 2 0 + sin 2 0 


a) 

( 2 ) 


because cos (- 0) = cos 0 and sin (- 0) = — sin 0. On using the theorem 
of Section 116 in the^numerator of (2), we obtain 


r(cos a + i sm a) r r , 

-7- - a . • — sc = - [cos (a - 0) + i sm (a - 0) J. 

s(cos 0 + i sm 0) s L 

EXERCISE 67 

Find the quotient in trigonometric form: 


(3) 


1 . 


2 . 


3. 


4. 


7 (cos 120° + i sin 120°) 

4 (cos 75° + i sin 75°) 

12(cos 275° + i sin 275°) 
4 (cos 180° + i sin 180°) ’ 

25(cos 250° + i sin 250°) 
5(cos 380° + i sin 380°) ' 

243(cos 160° + i sin 160°) 


5. 

6 . 

7. 

8 . 


6(cos 230° 

+ i sin 230°) 

V2 

+ iV 2 

2 - 

■ 2iV3 

3(cos 150° 

+ i sin 150°) 

15 (cos 80° 

+ i sin 80°) 

2 (cos 40° 

— i sin 40°) 

5(cos 125° 

- i sin 125°) 

2(cos 80° 

— i sin 80°) 


V3 - i 

9. Prove that the reciprocal of (cos d + i sin d) is 
[cos (— 6) + i sin (— 0)]. 

10. If k is any positive integer and H is any complex number, let us 

define H~ k by the equation H~ k = • Prove that 

[r(cos a + i sin a)] -fc = r -fc [cos (— ka) + i sin (— fca)]. 

Notice that this proves that the statement of De Moivre’s theorem remains 
true if the exponent n involved is any negative integer* n = — k. 

11. Expand (cos 9 + i sin 0) 5 by De Moivre’s theorem. Also, expand 
the expression by use of the binomial theorem. By equating the two 
results, obtain expressions for cos 59 and for sin 59 in terms of sin 9 and cos 9. 

12. Repeat Problem 11 for the case of (cos 9 + i sin 0) 7 . 

* For fractional powers, see Appendix, Note 7. 














GENERAL REVIEW 


169 


EXERCISE 68 

Except where otherwise specified , employ jour-place or five-place logarithms 
in any problem where it appears that they would be useful. 

1. Sketch an angle of 125° in its standard position on a coordinate 
system. Give the values of one negative and one positive coterminal angle. 
Indicate the three angles by curved arrows. 

2. Construct the acute angle a and find all of its functions if 
(a) tan a = A; (6) sec a = 

3. Prove that if d is in quadrant III, then tan 6 and cot 6 are positive 
and all other functions of 0 are negative. 

Find all junctions oj 0 if the terminal side of 6, in its standard position , 
goes through the given point: 

4. (- 3, 4). 5. (15, - 8). 6. (0, 3). 7. (- 2, 0). 8. (2, 5). 

9. Determine the functions of 45° by use of a right triangle with one 
leg 2 units long. 

10. Determine the functions of 30° and of 60° by use of a right triangle 
whose hypotenuse is 6 units long. 

Find each function by use of Table VII: 

11. tan 62° 37'. 13. sin 130°. 15. tan 243°. -17. cos (- 115°). 

12. sin 79° 33'. 14. tan 160°. 16. sin (- 15°). 18. sec 623°. 

19. Describe what is meant by writing tan 270° = °o. 

20. Find the acute angle a to the nearest minute by interpolation in 

some table: (a) tan a = 1.706; (6) cos a = .4138; (c) sin a = .9705. 

Find all functions of the angle without using a table: 

21. 210°. 22. 150°. 23. - 135°. 24. 315°. 25. - 120°. 26. - 225°. 

27. Select a point on the terminal side of the angle, in standard position, 

and determine its functions by use of their definitions: (a) 120°; (6) 225°. 

Solve right triangle ABC of page 15 without logarithms; use Table VII: 

28. c = 17.5; a = 10.5. 29. a = 42° 36'; b = .45. 30. c = 63; a: = 6° 35'. 

31. Solve right triangle ABC of page 15 without logarithms by inter¬ 
polation in Table XI if /S = 69° 16.7' and a = 250. 

Problems 32 to 35. Solve Problems 28 to 31, inclusive, by use of four- 
place or five-place logarithms and check the solutions. 

36. For a right triangle ABC, with the right angle at C, write a formula 
(1) for a in terms of j8 and b; (2) for finding a by use of a and c; (3) for 
c in terms of b and a. 


170 


TRIGONOMETRY 


Solve each equation: 


37. sin x = 

38. tanx = 1. 

39. cot x = V3. 

40. cos x = — 

41. cos x = — JV^2. 

42. sin x = — |V3. 

43. sec x = — 2. 

44. esc x = v 2. 


45. cos x = .8192. 

46. sin x = .4899. 

47. tan x = - .4734. 

48. cot x = - 1.492. 

49. cot 2 x = 1. 

50. sin 2 x = 1. 

51. 2 cos 2 x = 5 cos x. 

52. sin 2 x = 2 sin x. 


53. cot x + sin x cot x = 0. 

54. 2 sin 2 x + sin x = 1. 

55. 2 esc x + 3 = 2 sin x. 

56. 2 cos 2 x — 2 sin 2 x = 3. 

57. tan 2 x + sec 2 x = 7. 

58. sin 3x = 1. 

59. cos 2x = 

60. tan2x = — 1. 


Prove each identity by use of the definitions of the functions: 


61. tan 9 cos 6 = sin 9. 

62. sec 2 9 - 1 = tan 2 9. 


tan 2 9 cot 2 9 
sec 2 9 ^ esc 2 9 


64. tan 9 = 


sec 9 
esc 9 


65. Plot a point whose abscissa is - 3 and ordinate is 4. Compute the 
radius vector of the point. 

66. Without using a table or a figure, find all functions of 9 if 9 is in 
quadrant III and tan 9 = k i~. 


Change to a form involving only sin 9 and cos 9 and simplify: 


sin 9 + tan 9 
1 + sec 9 


68 . 


sec 9 

cot 9 + tan 9 


69. 


tan 9 — 1 
tan 0 + 1 


Obtain the number N whose logarithm is given: 

70. log 3 N = 4. 71. log 5 N = 2. 72. log 2 IV - — 4. 73. logi 6 N = b 

74. Define the logarithm of N to the base 10. 

75. Write an equivalent logarithmic equation by use of the definition 

of a logarithm: (a) N = 17 5 ; ( b ) N = 10“ 667 ; (c) N = 5 1 - 32 . 

76. By direct use of the definition of a logarithm, prove that 
loga \_{MP) -J- NQ2 = loga M + log„ P - log a N - lo go Q. 

77. Each number is the common logarithm of a number N; tell the 
characteristic and mantissa of log N: (a) 3.895; (6) — 3.0567; (c) — 6.1495. 


Compute by use of four-place or five-place logarithms: 


78. (2.68) 3 . 1 

79. v^.00314. . ‘ 38.93 



82. 


21.9567 

86.7353' 


9.32(.0418) 
65(— .152613) 


I 268.356 
\ 573(.249)' 


85. 


.739 

432v / .0185 


86. Find the length of arc subtended in a circle of radius 10 inches by a 
central angle of 3.7 radians. 















GENERAL REVIEW 171 

87. How many radians are there in a central angle in a circle whose 
radius is 15 inches if the angle subtends an arc 45 inches long? 

88. Sketch a graph of each of the trigonometric functions of x from 
x = — 7r to z = 7r by use of only a few points. From each graph, describe 
how the function varies as x increases from — 7r to t. 


Prove each identity: 


89. 


90. 


91. 


sec x — cos x 
esc x 


= sin 2 x tan x. 


92. 


esc y — sec y _ cos y — sin y 
esc y + sec y cos y + sin y 


cot 2 x + 1 
tan 2 x + 1 
sin 2 x 
1 — cos x 


cot 2 x. 

1 + cos x. 


93. 


1 + COS X = 


sin x 

esc x — cot x 


94. 


cot x — sin x tan x 
sec x 


cos x 
tan x 


sin x 
esc x 


95. Change to radian measure: (a) 325°; ( b ) 160° 35'. 

96. Change to degree measure: (a) 1.5 radians; (6) radiant. 

97. What is meant by saying that sin 9 is a periodic function of 0? 
What is the smallest period of sin 0? Tell another period of sin 9. 

98. The bearing of a point P as seen from A is N 26° 37' E. How far 
north and how far east is P from A if PA = 36,723 feet? 

99. If P is 650 yards south and 316 yards west of A, find the bearing 
of A as seen from P. 


Without tables, find the indicated function by using functions of the specified 
angles. Results may be left in radical form. 

100. cos 165°; use 45° and 120°. 103. sin 75°; use 150°. 

101. sin 285°; use 330° and 45°. 104. cos 67^°; use 135°. 

102. tan 75°; use 120° and 45°. 105. cot 1121°; use 225°. 

106. sin 60°; use 30°. 107. tan 240°; use 120°. 108. cos 120°; use 60°. 

109. Without tables, find the functions of (a + /3) and of (a — j8) if 
sin a = -£t and cos /3 = — if a is in quadrant II and /3 is in quadrant III. 


Prove the identity: 


110 . 


2 cot 9 
tan 29 


= cot 2 0 — 1. 


111 . 

112 . 

116. 


sec 29 = 
esc 2x = 


1 

cos 2 9 — sin 2 9 

CSC 2 X 

2 cot x 


sin (x - 30°) - cos (120° - x) = 0. 


113. 


sec 2x = 


sec 2 x 
1 — tan 2 x 


114. 


sin 5x — sin 3a; 
cos 5x + cos 3x 


= tan x. 


115. tan (x — 45°) = j C °^ - • 
v 1 + cot x 

117. sin 3x = 3 sin x — 4 sin 3 x. 

















172 


TRIGONOMETRY 


Solve each equation: 

118. cos 2x + cos x = 0. 121. cos 4x + cos 2x = cos x. 

119. sin 2 a: — 2 cos x = 0 . 122 . sin \x = - h 

120. sin 5x — sin 3a: = 0. 123. cos %x = 2. 

124. Sketch a diagram like Figure 69, page 102, employing different 
letters, and prove the addition formulas for the sine and cosine. 

125. For triangle ABC, page 120, write a formula (1) for b in terms of 
a, a, and /3; (2) for finding a by use of b, c, and a; (3) for finding 0 by use 
of a, b, and a; ( 4 ) for finding a by use of a, b, and c (give two formulas); 
( 5 ) for the area in terms of o, b, and 7 ; ( 6 ) for the area in terms of a, b, 
and c. 

Solve triangle ABC, page 120, completely, except where only one part is 
requested; check where a complete solution is asked for. Use either four-place 
logarithms or four-place natural functions, whichever is most convenient. 

126. a*=50; 0 = 53°; a = 67°: find 6 . 129. a = 6 ; 6 = 5; c = 4. 

127. c = 35; 6 = 30; a = 25: findy. 130. a = 63.8;6 = 96.4;0 = 128°17'. 

128. o = 25; 6 = 30; a=35°: find/3. 131. a = .1568;6 = .3349 ;7 = 69° 18'. 

132. Solve triangle ABC, page 120, by use of five-place logarithms: 
o = 1.95; a = 61° 42.3'; 0 = 52° 19.7'. 

133. Without using logarithms, find the angle of elevation of the sun 
when a tree 180 feet high casts a horizontal shadow 240 feet long. 

Express each function in terms of a function of an acute angle less than 45°; 


134. 

sin 73°. 

137. 

CSC 253°. 

140. 

sin ( — 

153°). 

143. 

sin 645°. 

135. 

tan 142°. 

138. 

cos 152°. 

141. 

cos ( — 

27°). 

144. 

cot 708°. 

136. 

sec 310°. 

139. 

cot 118°. 

142. 

cot ( — 

220°). 

145. 

esc (- 895°). 


Express each function in terms of a function of x: 

146. sin (270° + *). 148. cot (90° + as). 150. sec (2ir - x). 

147. cos (180° + x). 149. tan (fx + x). 151. esc (—a;). 

152. Let 0 be an obtuse angle and let /3 = 0 — 90°. Place 0 and 0 in 
their standard positions on a coordinate system and from the figure prove 
the reduction formulas for functions of (0 — 90°). 

153. From the top of a cliff, the angles of depression of two successive 
mile posts on a horizontal road running due north are 74° 15' and 25° 36', 
respectively. Find the elevation of the cliff above the road. 

154. Find the length of the longest diagonal of a parallelogram whose 
sides are 175.3 and 286.5 feet long if one angle is 133° 22'. 

155. In a certain regular octagon the length of a side is 157.6 inches. 
Find the radii of the inscribed and circumscribed circles. 


GENERAL REVIEW 173 

156. A tower at the top of an embankment casts a shadow 125.6 feet 
long, straight down one side, when the angle of elevation of the sun is 
48° 25'. If the side of the embankment is inclined 33° 26' from the hori¬ 
zontal, find the height of the tower. 

157. The lengths of two adjacent sides of a quadrilateral are 150 feet 
and 268 feet, respectively, and the angle between the sides is 67° 46'. The 
other sides of the quadrilateral are respectively perpendicular to the speci¬ 
fied adjacent sides. Outline a method for finding the unknown sides and 
angles of the quadrilateral. 


PROBLEMS REFERRING TO CHAPTERS X AND XI 


158. If (r = 3, 9 = 150°) are polar coordinates of a point, give three 
other pairs of polar coordinates for it, with r negative in two pairs and 9 
negative in at least one pair. 

159. Find the rectangular coordinates of the point whose polar coordi¬ 
nates are given: (a) (r «* 3, # - 150°); (6) (r = 5, 9 = 

160. Find a set of polar coordinates for the point whose rectangular 
coordinates are given: (a) (- 1, - 1); (6) (- Vs, 1); ( c ) (- 5, 0). 


161. Give two positive or zero values and one negative value for each 

inverse function: (a) arcsin 1; (6) arccos (— (c) arccsc -?=• 

\ 2 / V3 

/ y/2\ 

162. Find the principal value of arccos (-— 1• 


Find the value of the expression: 

163. Arcsin |. 165. Arcsin (—1). 167. sin arcsin (— f). 

164. Arctan (—1). 166. cos arccos i. 168. sec Arctan (— 1), 


SUPPLEMENTARY PROBLEMS 

Prove each identity. If radicals occur, suppose 0° ^ x ^ 90°. 


169. tan£ 
A 


170. tan x = 

171. 


172. 


1 — COS X 

sin x 

1 — cos 2x 


sin 2x 

1 — tan x 1 — sin 2x 


1 + tan x 


V 


CSC x — 1 
CSC X + 1 


cos 2x 

1 — sin x 
cos x 


173. 

174. 

175. 

176. 


1 


cos x — 1 
cos 6a; — 1 
cos 6x + 1 
1 


2 cos x + 1 
cos 2x — cos x 

= 1 — sec 2 3x. 

1 — cos x 


cot X -f- CSC X 


sin x 


cos x , sin re 0 

-■:-- 7 - = COS 3x. 

sec 4a; esc 4a; 


177. cos 2a; + 2 sin 2 3a; — cos 4a; = 4 sin 3a; sin 2a; cos x. 

178. sin 5a; + sin 3a; — sin 2a; = 4 cos x cos y sin 
















174 


TRIGONOMETRY 


Solve each equation: 

179. cot 4z + cot 2x 

180. tan 3x — tan x 

181. cos x - sin 

182. 2 sin x - V^2 c( 


= 0 . 

183. 

= sec 3x. 

184. 

x = 0. 

185. 

>s x = 0. 

186. 


4 cos 2x = 5. 

5 tan 3x — 2 = 0. 
cos x + 1 = sin x. 
esc x — cot x = 1. 


Compute the expression; assume that x > 0 and y > 0: 

187. sin [90° + Arccos (- *)]. 189. tan (tt + Arcsin fV). 

188. sin (Arccos x — Arcsin y ). 190. cot Arcsin x. 

191. Graph y = sin | from x = 0 to x = 4t. 

192. Graph y = sin 2x — cos x from x = 0 to x = 2tt. 

193. Find the unknown base: (1) logo 8 = 3; (2) log„ 125 = f. 

194. Find the specified logarithm: (1) log 6 216; (2) logioo .0001. 

195. An object at P is acted upon simultaneously by the following 
horizontal forces in the specified directions: 168 pounds, A 38 15 ' E) 
157 pounds, N 26° 42' W. Find the magnitude and direction of the re¬ 
sultant force. Use Table I and five-place logarithms. 

196. Find the vertical and horizontal components of a force of 216 
pounds acting downward and inclined 18° 16' from the horizontal. 

197. Find the force which is exactly sufficient to keep a 500-pound 
weight from sliding down a plane inclined 26° 57' from the horizontal. 

198. An object at P is acted upon simultaneously by two forces of 157.6 
pounds and 384.3 pounds. If one of the angles between the directions of 
the forces is 78° 32', find the magnitude of the resultant force and the acute 
angle between its direction and that of the force of 157.6 pounds. 

199. A flywheel 36 inches in diameter is driven by a belt moving over 
the rim with a velocity of 3000 feet per minute. Find the angular velocity 
of a point on the rim of the wheel in radians per second. 

200. A wheel 2 feet in diameter makes 250 revolutions per minute. 
Find the angular velocity, in radians, and the linear velocity per minute 
of a point on the rim of the wheel. 

201. Find tan 0° 27.4' by use of Table X6, and also by use of Table Xa. 

202. Find the angle by use of Table X6: log sin a = 7.84532 — 10. 

203. Solve for x: 5(1.03) 2 * = 15.731. 

204. Compute (1.05) 120 with an error of not more than one unit in the 
fifth significant place. 

205. Without using Table XIV, compute the natural logarithm of 5.07. 
Verify the result by use of Table XIV. 


APPENDIX 
Note 1 


A logarithmic scale. In Figure 103, after selecting the unit of length, 
HK, we lay off a scale above the line in the usual way. Below the line, 
each value of x is placed at the point under the value of X which equals 
log x (where “log ” means “ logi 0 ”); that is, if x is under X, then log x — X. 
Illustration 1. Since log 1 = 0, x = 1 is under X = 0. 

Since log 4 = 0.6, approximately, x = 4 is under X = .6. 

H (unit of length) K 

—.4 -.2 O .2 .6 .8 1 1.4 1.6 2 2.2 

H i i 1 1 i 11 1 i i 1 i i 1 it i i i S I j l i i i I 1 i r~ T 

.4 .6 .6 .8 1 2 3 4 6 6 8 10 20 30 40 60 80 100 200 300 ^ 

Fig. 103 


If we blot out the upper numbers in Figure 103, we obtain the scale in 
Figure 104, called a logarithmic scale. For contrast, the upper scale in 
Figure 103 may be called a uniform scale. On a logarithmic scale, if the 
unit of distance is properly chosen, the distance from x = 1 to x = N equals 
log N. Since N has no real logarithm if N ^ 0, we find, neither zero nor 
negative numbers on a logarithmic scale. 


H (unit of length) K 


l i i l i 11 

.4 .5 .6 .8 1 


I I M i l " 

5 6 8 10 


20 30 40 


I I I I I I 
60 80 100 


Fig. 104 


Note 1. Recall that, if a is any specified base for a system of logarithms, then 
logio N = (logio a) (log a N). Hence, if the distance from x = 1 to x = a on the 
scale in Figure 104 is selected as the positive unit of length, then the distance 
from x = 1 to x = N on the scale equals logo N. Thus, a logarithmic scale made 
by use of any specified system of logarithms may be used as the logarithmic 
scale with any other system. 

Note 2. Logarithmic scales are employed on the various scales on a slide 
rule.* The usual 10-inch slide rule permits rapid computation with about the 
accuracy obtained by using three-place logarithms. The longer the slide rule, the 
greater is the accuracy obtainable by using it. The fundamental principle involved 
in the primary use of a slide rule is illustrated in Figure 105. Scales A and B 

* The detailed use of a slide rule is explained in the circular obtained on purchasing 
any of the standard rules. 


175 







176 


TRIGONOMETRY 


are logarithmic. To find, for instance, the product 3 X 4, we place the 1 of scale B 
under the 3 of A, Then, above the 4 of B we read the product 3 X 4 on A. 
We obtain this result because OK = OH + HK = log 3 + log 4 = log (3 X 4). 

AJ_ ? » f 6 ? 7 8 9 10 11 12 .20_ 

~o p * 

, _i_I-L_-.-1-L- 

1 2 3 4 5 6 7 


Fig. 105 
Note 2 

Angles near to 0° or 90°. The tabular differences increase rapidly in 
Table IX as we approach 0° in the columns headed log sin, log tan, and 
log cot. Hence, if a logarithm is determined by interpolation in these 
columns the error is extremely large on page 36 of Table IX and remains 
undesirably large (> .00001) until we reach page 40. Table X is provided 
for avoiding these large errors of interpolation. 

Let —= s and ^ a !\ — = t, where M is the number of minutes in a. 
M M 

Then, sin a = sM and tan a = tM; 

log sin a = log s + log M; log tan a = log t + log M. 

If we let S = log s and T = log t, then 

log sin a = S + log M\ log tan a = T + log M. (1) 

S and T are computed by the methods of advanced mathematics; the 
values of S and T are tabulated in Table Xa for a between 0° and 3°. 

A. To find log sin a , log tan a, or log cot a for a near to 0°: 

1. Express a in minutes and decimal parts of a minute, to find the value of M. 

2. Find log M from Table VIII and S and T from Table Xa. 

3. Use (1) to find log sin a and log tan a; to find log cot a, recall that 

cot a = —-— > or log cot a = — log tan a. 
tan a 

B. To find log cos /3, log tan j8, or log cot 0 for j8 near to 90°, let 
a — 90° — /3. Recall that cos = sin a, tan j3 = cot a, and cot j8 = tan a; 
then use (A). 

Example 1. Find log sin 1° 26' 12". 

Solution. 1. = .20; M = 86.20'. Hence, S = 6.46368 - 10, from Table Xa. 

2. From Table VIII, log 86.20 = 1.93551. Hence, 

log sin 1° 26' 12" = 1.93551 + 6.46368 - 10 = 8.39919 - 10. 

Comment. The preceding method is accurate for angles expressed to the 
nearest second except for errors due to rounding off in the last place. 







APPENDIX 177 

C. To find an angle a near to 0° when log sin a (or log tan a ) is given: 

1. Find a approximately to the nearest minute by inspection of Table IX. 

2. Find S (or T) from Table Xa; substitute for S and log sin a in (1), and find 
log M; then obtain M from Table VIII; usually the result will be in error by at 
most one second. 

Note 1. .If log cot a is given for a near to 0°, first find log tan a from 
log tan a = — log cot a, and then use (C). 

D. To find an angle j8 near to 90° if log cos /3, log tan /3, or log cot j8 
is given, write (3 = 90° — a, and first find a by use of (C). 

Illustration 1. If log cot /3 = 8.65246 — 10, then /3 is near to 90°; if 
a = 90° — /3, then log tan a = 8.65246 - 10. By use of (C) we could find a 
and then obtain (3 — 90° — a. 

Example 2. Find a if log sin a — 8.66345 — 10. 

Solution. 1. By inspection on page 38 of Table IX, we find that a is nearer to 
2° 38' than to 2° 39'. Hence, approximately, M = 158', and, from Table Xa, 
S = 6.46357 - 10. 

2. In (1), substitute for log sin a and S: 

8.66345 - 10 = 6.46357 - 10 + log M ; 
log M = 8.66345 - 6.46357 = 2.19988. 

From Table VIII, M = 158.44', or a = 2° 38.44' = 2° 38' 26". 

Note 2. A second method for solving problems like Examples 1 and 2 is 
provided by Table Xfe, for angles very near to 0°. If angles are expressed only 
to the nearest .1', Table Xb demands no interpolation. If angles are expressed to 
the nearest 1" or .01', Table Xb requires interpolation, which sometimes is subject 
to greater error than arises through the methods of Examples 1 and 2. Except 
for very refined work, the use of Table Xb without interpolation should be satis¬ 
factory. 

Note 3. A slowly changing function of an unknown angle a offers a poor 
means for determining a. Thus, if we are given log cos a = 9.99985 — 10, we 
find from Table IX, page 37, that a may have any value from 1° 29' to 1° 31', 
inclusive, because log cos a changes very slowly. Hence, when possible, we should 
avoid using cos a in determining a if a is near to 0°, or sin a if a is near to 90°. 

Note 3 

Non-geometrical proofs of general reduction formulas. On the basis of 
Sections 55 and 57, all other reduction formulas can be proved analytically 
by the method indicated in the following discussion. 

Illustration 1. By use of Section 57 and then Section 55, 

cos (90° — 0) = cos [90° + (- 0)] = sin (- 0) = — sin 0. 

Illustration 2. By use of Section 55 and then Section 57, 

sin (- 0 - 90°) = sin [- (90° + 0)] = - sin (90° + 0) = - cos 0. 


178 


TRIGONOMETRY 


Similarly, as in Illustrations 1 and 2, we may prove that any function of 
(90° — 0), (0 — 90°), or (— 0 — 90°), is numerically equal to the cofunc¬ 
tion of 0. These facts, together with Section 57, justify the following 
statement: 


I. The addition of 90°, or of - 90°, to 6 or -6 gives an angle each of 
whose functions is numerically equal to the cofunction of 6. 

By successive applications of (I) we can prove the facts stated in the 
summary of Section 59. 


Illustration 3. By three applications of (I) we obtain 
sin (270° - 0) = ± cos (180° - 6) [Because (270° - 0) = 90° + (180° - 0)] 
= ± sin (90° - 0) [Because (180° - 0) = 90° + (90° - 6)] 
= ± cos 9. (1) 

In passing from (270° — 9) to 9, three subtractions of 90° occurred; hence, by 
use of (I) we had to change from function to cofunction three times. Since three 
is an odd number, the final function, cosine in (1), is the cofunction of the original 
function, sine. 


Illustration 4. Consider the functions of (n • 90° + 9) where n is a positive 
integer. By n applications of (I) we obtain 

sin (n • 90° + 9) {Because n • 90° + 9 = 90° + [(ra — 1) • 90° + 0]} 

= ± cos [(w — 1) • 90° + 9~] 

{Because (n - 1) • 90° + 9 = 90° + [(» - 2) • 90° + 0]} 


= ± sin [(n - 2) • 90° + 0] 

= =b cos [(» - 3) • 90° + 0] = 


± 


sin 9, if n is even 
cos 9, if n is odd 


( 2 ) 


We arrived at the final result in (2) by n steps, in each of which we subtracted 
90° from the preceding angle and therefore changed from function to cofunction. 
Hence, the final function has the same name, or else is the cofunction of the given 
function, sine, according as n is even or odd. Similarly, it follows that any func¬ 
tion of (n • 90° + 9) is numerically equal to the same function, or to the cofunc¬ 
tion of 9, according as n is even or odd. 


Note 4 

Line values of the functions. Let 9 be any angle; place 9 in its standard 
position on a coordinate system. Construct a circle with radius 1 and 
center at the origin, which intersects the positive halves of the x-axis 
and y- axis at U and V, respectively. Construct tangents to the circle at U 
and V. Let P, T, and C be the points where the terminal side of 9, or 
this side extended through 0, intersects the circle and its tangents at U 
and V, respectively. Then, in Figures 106 and 107, OA and AP represent 
the coordinates of P. 

In the resulting diagram we agree that all line segments will be con¬ 
sidered as directed segments, and that the unit of length will be that 



APPENDIX 


179 



Y 



which is used for the coordinates. A segment parallel to a coordinate axis 
will be considered positive or negative according as the direction is the 
same as or opposite to the positive direction on the axis. On OP we 

define from 0 to P as the positive direction. Then,* 

sin 0 = AP; tan 0 = UT\ sec 0 = OT] . . 

cos 0 = OA ; cot 0 = VC] esc 0 = OC. 

Illustration 1. In Figure 107, OT is negative and hence OT = — OT, where 
OT is the positive number of units of length in OT. Let a be the acute angle XOT. 

Tyr _ ' _ 

Then, sec a = = OT because OU = 1. But sec 0 = — sec a = — OT = OT. 

Hence, if 9 is in quadrant II, sec 9 = OT. 

Note 1. The terminal side of 9 may fall in any one of the four quadrants, 
or on either half of either axis. For each of these possible cases, it would be 
necessary to give a separate proof, like that of Illustration 1, for each of equa¬ 
tions 1. 

Note 5 

Another method for proving a trigonometric identity. A trigonometric 
identity may sometimes be established as follows: 

1. Assume that the conjectured identity is true and manipulate it until an obvious 
identity is obtained. 

2. Start anew with this final obvious identity and prove that, from it, the conjectured 
identity can be obtained. 

The method of this note is dangerous in inexperienced hands because 
special ability is sometimes required in deciding whether or not the details 
in Step 1 may be reversed, as is necessary in Step 2. The method is not 
recommended for use in this course because of the dangers in connection 
with Step 2 and, also, because of the nature of the applications of trigono- 

* When 0 has a value for which some function of 6 is infinite, the corresponding 
equation in (1) is meaningless. 













TRIGONOMETRY 


180 

metric identities in more advanced mathematics. These applications 
usually require that the methods of Section 74 should be used. 

...... 1 + cos x sin x 

Example 1. Prove the identity: • ~— = x _ cos ^‘ 


Solution. 1. Assume that the identity is true and multiply both of its sides 
by sin x{\ — cos x ): 


or, 


(1 + cos m) (1 - cos x) = sin 2 x; 
1 — cos 2 x = sin 2 x. 


( 1 ) 


We recognize that (1) is true because sin 2 x + cos 2 x = 1. 

2. If x has any value such that sin x{\ — cos x) ^ 0, we may divide both 
sides of (1) by sin x(l - cos x). This division gives 

1 - cos 2 x _ sin 2 x 1 + cos x = sin x 

sin x(l — cos x) sin x(l — cos x) sin x 1 — cos x 


Hence, the given identity has been proved because we have shown that it is true 
if (1) is true. 

Comment. The preceding solution would be absolutely unacceptable if 
Step 2 were omitted. 


Note 6 


General proof of the addition formulas. Suppose that the following 


formulas have* been proved geometrically in case a and /3 are between 
0° and 90°, inclusive: 

sin (a + j8) = sin a cos /3 + cos a sin /3; (1) 

cos (a + j8) = cos a cos /3 — sin a sin /3. (2) 

Theorem I. 7/ a and j3 are any two angles for which (1) and (2) are true, 
then they remain true if either a or (3 is increased or decreased by 90°. 

Proof. 1. Let a' = a + 90°. Then sin a' = cos a; cos a' = — sin a. 

2. sin (a' + j8) = sin [(a + /3) + 90°3 

= cos (a + (3) [Section 57, page 69] 

[Using (2)] = cos a cos j8 — sin a sin /3. (3) 

3. Replace a by a' on the right in (1) and use Step 1: 

sin a r cos + cos a' sin /? = cos a cos j8 — sin a sin /3. (4) 

Since (3) and the right side of (4) are identical, hence 

sin (a' + /3) = sin a' cos j(3 + cos a' sin /3. (5) 


Since a' = a + 90°, (5) states that (1) is true if a is increased by 90°. 

4. Similarly, by starting with a' = a — 90°, it can be proved that (1) 
is true if a is decreased by 90°. Similar facts can be proved about (2). 
To prove the theorem as it applies to /S, we would similarly use (S' = /3 + 90° 
and 13' = 13 - 90°. 








APPENDIX 


181 


Theorem II. Formulas 1 and 2 hold for all values of a and (5. 

Proof. 1. Let y and 0 be any angles not between 0° and 90°, inclusive. 
Then, there exist angles a and /3, between 0° and 90°, inclusive, such that 
7 = ot m • 90° and 0 = /3 + n • 90°, where m and n are appropriate 
integers. We know that (1) and ( 2 ) are true for a and /3. 

2 . We can pass from a to 7 through a sequence of angles, by successive 
additions of 90°, if m > 0, or subtractions of 90°, if m < 0. By Theorem I, 
formulas 1 and 2 remain true if a is replaced in turn by the successive angles 
of this sequence. Hence, finally, we may replace cc by 7 in ( 1 ) and ( 2 ). 
Similarly, it follows that we may replace /3 by 0 in (1) and ( 2 ). Therefore, 
(1) and (2) are true if a and /3 are replaced by 7 and 0, respectively. 


Note 7 

De Moivre’s theorem for fractional powers. The agreements of ele¬ 
mentary algebra concerning a symbol like Hn apply only when H is real 
and when some nth root of H is real.* Suppose that we now make the 
following agreement:! for any complex number H = R (cos a + i sin a), 
and any positive integer n, Hn represents that nth root of H obtained by placing 
k = 0 in formula 4, page 166: 

H » = [i?(cos a + i sin a)]» = 2?^ cos ^ + 1 sin • (1) 


With this agreement, equation 1 states that De Moivre’s theorem on 
page 165 is true if the exponent is changed to 1/n. If in (1) we change 
a to (a + 360°), then to (a + 720°), • • •, etc., we obtain the other roots 
given in (4), page 166. Hence, that formula is equivalent to De Moivre’s 
theorem for the exponent 1 /n. 

If m is any positive integer, let Hn = ( Hn) m . Then, on using De Moivre’s 
theorem for an mth power, from ( 1 ) we obtain 


H- = [ffi(cos l + i sin 2 )]“ = «;(cos ^ + i sin 


( 2 ) 


Let H-fi = — • Then, from (2), 

Hn 


H-* - ~ 


IT? ma . . . ma\ 

M 71 Rnl cos-h^sin — ) 

\ n n / 


(3) 


* See page 63, Hart’s College Algebra, Alternate Edition. D. C. Heath and Company, 
publishers. 

t This is not a universal agreement. In any application it is always essential 
1 

to state which root is meant by Hn. 



182 TRIGONOMETRY 

From (3), by use of Problem 9 on page 168, we obtain 

[fl(cos a + i sin = # _ ^cos + i sin ^ ‘ (4) 

Equations 2 and 4 state that De Moivre’s theorem is true if the ex¬ 
ponent is p/q where p and q may be any positive or negative integers. 


Note 8 


General proof of the law of cosines. 1. Let a be any side of triangle 
ABC. Place the triangle on a coordinate system with A at the origin, 
B on the positive x-axis, and C above the z-axis. Drop a perpendicular 
CD from C to the z-axis. Let DC = h and AD = m. Recall that a line 
segment parallel to a coordinate axis is considered as positive or negative 
according as the direction of the segment is the same as or opposite to the 
positive direction on the axis. 

2. The angle a is in its standard position on the coordinate system and 
C is a point on the terminal side of a; the abscissa of C is x = m and the 
radius vector of C is r = 6. 


3. For any triangle ABC, 

, v cos a 


m 

V 


m = b cos a. 



Fig. 108 

Substitute (1) in (4): 


4. From the figure, 

DB = AB — AD = c — m. 

5. By use of the Pythagorean theorem, 
h 2 = b 2 — m 2 ; a 2 = h 2 + (DB) 2 . 

From (2) and (3), 

a 2 = b 2 - m 2 + (c - m) 2 = b 2 + c 2 - 2 me. 
a 2 ~ b 2 4“ c 2 — 26c cos a. 


( 1 ) 

( 2 ) 

(3) 

(4) 


Note 9 

Terminology relating to surveying. In the routine applications of sur¬ 
veying, it is assumed without appreciable error that the theoretical surface 
of the earth, with respect to which we measure elevations, is a plane 
instead of a sphere. This ideal plane is referred to as the horizontal plane. 

Let M and P be two points on the earth, perhaps at different elevations. 
Then, the horizontal projection of the directed line segment MP is called 
the course of MP or, simply, course MP. The acute angle a made by 
MP with the horizontal is called the inclination of MP. From page 51, 
it follows that, if MP represents the length of MP, 

course MP = MP X (cosine of inclination); (1) 

(vertical projection of MP) = MP X (sine of inclination). (2) 





APPENDIX 


183 


It is easily verified from a figure that the inclination, a, oi MP satisfies 
the equation 

vertical projection of MP 

tan a = --—-—— ^- (3) 

course MP v ' 


A negative sign is prefixed to the inclination and to the horizontal projec¬ 
tion of MP in case P is below M. 


Note 1. In the legal description of any boundary line MP of a parcel of ground, 
such as a city lot, the specified length is course MP and not MP itself. 

Example 1. A surveyor goes from M to R by the broken line path, 
MP, PQ, and QR where M, P, Q, and R lie in the same vertical plane. 
Find the length of MR, its inclination, and course MR, if MP, PQ, and 
QR have the lengths and inclinations given in the following table. 

Solution. 1. (The student should construct a figure.) We compute the courses 
and vertical projections of MP, PQ, and QR by use of (1), (2), and Table VI-I. 
Thus, course PQ = 300 cos 7° 30' = 297.4 yd. 

The results are’summarized in the following table. 


Line 

Inclina¬ 

tion 

Measured 

Length 

Course 
of Line 

Vertical Projection 

MP 

15° 20' 

200 yd. 

192.9 yd. 

(+) 52.9 yd. 


PQ 

(-) 7° 30' 

300 yd. 

297.4 yd. 


( —) 39.1 yd. 

QR 

9° 10' 

500 yd. 

493.6 yd. 

(+) 79.6 yd. 


MR 



983.9 yd. 

( + ) 93.4 yd. 



2. Course MR = course MP + course PQ + course QR. Similarly, we find 
the vertical projection of MR. To find the inclination of MR, we use (3): 

tan a = <2 = 5° 25'. (From Table VI) 


3. 


To find MR, the length of MR, we use 


cos a 


course MR _ 
MR ’ 


MR 


983.9 
cos 5° 25' 


988 yd. (Using logarithms) 


Comment. In Example 1, we would call MR the closing line of the surveyor’s 
path, and course MR the closing course. 

In referring to the direction of MP, the surveyor means the direction of 
course MP ; this is the direction obtained by his measurements. 

The projection of course MP on the north-south line is called the 
latitude of the course, and is labeled positive or negative according as P 
is north of M or south of M. The projection of course MP on the east- 
west line is called the departure of the course, and is labeled positive or 
negative according as P is east of M or west of M. From page 51 it fol¬ 
lows that, if |S is the bearing angle of MP, 






















184 


TRIGONOMETRY 


latitude = ± (length of course) X (cosine of bearing angle); (4) 

departure = ± (length of course) X (sine of bearing angle); (5) 

numerical value of departure 


tan j3 = 


(6) 


numerical value of latitude 

Example 2. Find the bearing and length of the closing course MR if a 
surveyor follows the path MPQR where courses MR, PQ, and QR have 
the lengths and bearings given in the table below. 

Comment. Example 2 is similar to Problems 17 to 22, page 52. 

Solution. (The student should construct a figure.) The solution is summa¬ 
rized in the last two columns and last row of the following table. 


Course 

Bearing 

Length 

Latitude 

Departure 

MP 

N 32° 10' E 

200 yd. 

169.3 yd. 

106.5 yd. 

PQ 

N 22° 30' W 

400 yd. 

369.6 yd. 

- 153.1 yd. 

QR 

S 15° 50' W 

300 yd. 

— 288.6 yd. 

81.9 yd. 

MR 

N 27° 10' W 

281 yd. 

250.3 yd. 

— 128.5 yd. 


tan j8 = 


128.5 


j8 = 27° 10'. 


MR = = 281. 


250.3’ ^ " cos /3 

The direction of MR is N 27° 10' W. Tables XI, V, and VI were used. 


EXERCISE 69 

Given that B, C, and D lie in the same direction from A. Find the closing 
course AD, the length of AD, and its inclination: 


1 . 


Line 

Inclin. 

Length 

AB 

CO 

o 

CO 

00 

650 yd. 

BC 

6° 20' 

250 yd. 

CD 

- 5° 40' 

300 yd. 


Find the bearing and length of course AD: 


Course 

Bearing 

Length 

of 

Course 

AB 

N 21° 30' E 

250 yd. 

BC 

S 32° 15' W 

300 yd. 

CD 

N 14° 32' W 

325 yd. 


Line 

Inclin. 

Length 

AB 

- 6° 25' 

716 yd. 

BC 

8° 43' 

267 yd. 

CD 

- 4° 20' 

532 yd. 


Course 

Bearing 

Length 

of 

Course 

AB 

S 12° 43' E 

835 yd. 

BC 

N 20° 15' W 

672 yd. 

CD 

S 15° 26' E 

849 yd. 























































INDEX 


NUMBERS REFER TO PAGES 


Abscissa, 57. 

Absolute value, 56, 163. 

Addition formulas, 101. 
general proof of, 180. 

Amplitude, of a complex number, 163. 

Angle, acute, 2. 
functions of an, 2, 60. 
general definition of, 58. 

Angle of depression, 17. 

Angle of elevation, 17. 

Antilogarithm, 27. 

Approximate data, 9, 47. 

Area of a triangle, 140. 

Auxiliaries S and T, 176. 

Base of a system of logarithms, 21. 

Bearing of a line, 51. 

Characteristic of a logarithm, 24. 

Circumscribed circle, 147. 

Cofunction, 7. 

Cologarithm, 32. 

Common logarithms, 24. 

Complementary angles, 7. 
functions of, 7. 

Complex numbers, 160. 
conjugate, 161. 

De Moivre’s theorem for, 165. 

geometrical representation of, 162. 

multiplication of, 165. 

polar form of, 163. 

powers of, 165. 

roots of, 166. 

Conjugate complex numbers, 161. 

Cosecant, definition of, 2, 60. 

Cosine, definition of, 2, 60. 

Cosine of half an angle, 107. 
of a triangle, 137. 

Coterminal angles, 59. 

Course of a line, 182. 

De Moivre’s theorem, 165. 
for fractional powers, 181. 
for negative powers, 168. 

Departure of a course, 183. 


Difference of two angles, functions of 
the, 104. 

Directed line, 56. 

Double angle formulas, 107. 

Equations, trigonometric, 94, 115. 

Exponential equation, 38. 

Exponential function, 40. 

Extraneous solution, 97. 

Function, general definition of a, 1. 
trigonometric, 2, 60. 

Functions, of acute angles, 2. 
of angles near to 0° or 90°, 176. 
of any angle, 60. 
of complementary angles, 7. 
of double an angle, 107. 
of half an angle, 107. 
of the half angles of a triangle, 136. 
of the negative of an angle, 68. 
of (a + j8), 101, 103. 
of (a — (3), 104. 
of (± Q ± n ■ 90°), 70. 

Fundamental identities for functions of 
a single angle, 89. 

General angle, 58. 

Graphs, of the functions, 82. 
of the inverse functions, 157. 

Half-angle formulas, 107. 
for a triangle, 136, 137. 

Identities, 92, 112, 158, 179. 

•Imaginary number, 160. 

Inclination of a line, 182. 

Infinite values of functions, 79. 

Initial side of an angle, 58. 

Inscribed circle, 136. 

Interpolation, with logarithms, 29, 42. 
with natural functions, 10, 13. 

Inverse functions, 152. 
principal values of, 154. 

Irrational number, 20. 

Latitude of a course, 183. 

Law of cosines, 120. 
general proof of, 182. 


185 



186 


INDEX 


Law of sines, 124. 

Law of tangents, 132. 

Line values of functions, 178. 
Logarithmic equation, 38. 

Logarithmic scale, 175. 

Logarithms, characteristics of, 24. 
common or Briggs, 24. 
definition of, 21. 
mantissas of, 24. 
natural or Naperian, 24, 39. 
of the functions, 42. 
properties of, 23. 

Mantissa, 24. 

Modulus of a complex number, 163. 

of a system of logarithms, 40. 
Mollweide’s equations, 134. 

Natural or Naperian logarithms, 24, 39. 
Natural values of functions, 9, 43. 
Negative angles, 58. 

Numerical value, 56. 

Oblique triangles, 120. 
ambiguous case of, 127. 
analysis of solution of, 141. 
area of, 140. 

half-angle formulas for, 136. 
law of cosines for, 120. 
law of sines for, 124. 
law of tangents for, 132. 

Mollweide’s equations for, 134. 
Ordinate, 57. 

Periodicity of the functions, 78. 

Polar coordinates, 149. 

Principal values, 154. 

Products of sines or cosines, 110. 
Projection of a segment, 51. 

Quadrants, 57. 

Radian, 73. 

Radian measure, 73. 


Radius vector, 57, 149. 

Rational number, 20. 

Rectangular coordinates, 57. 

Reduction formulas, 68, 70, 177. 
Reduction to acute angles, 65, 70. 
Reference angle, 65. 

Refinement of results, 43. 

Regular polygon, 49. 

Resultant of two vectors, 53. 

Right triangles, formulas for, 15. 
solution of by natural functions, 15. 
solution of by logarithms, 45. 

Roots of a complex number, 166. 
Rounding off a number, 9. 

Secant, definition of, 2, 60. 

Significant digits, 9. 

Signs of the functions, 62. 

Sine, definition of, 2, 60. 

Sine of half an angle, 107. 

of a triangle, 137. 

Slide rule, 175. 

Standard position of an angle, 59. 

Sum of two angles, functions of, 101. 
Sum or difference of sines or cosines, 
111 . 

Surveying terminology, 51, 182. 

Tables, description for, 
logarithms, 27, 30. 
logarithms of the functions, 42. 
natural functions, 9, 13. 

Tangent, definition of. 2, 60. 

Tangent of half an angle, 107. 

of a triangle, 136. 

Terminal side of an angle, 58. 
Trigonometric equations, 94, 115. 
Trigonometric functions; see Functions. 

Variable, definition of a, 1. 

Variation of the functions, 78. 

Vector, 53. 

Vectorial angle, 149. 

Velocity, 75. 






ANSWERS TO EXERCISES* 


Exercise 1. Page 5 

In any problem which calls for all functions of an angle, the given answer states only 
the values of the sine, tangent, and secant, in this order; their reciprocals give the 
values of the cosecant, cotangent, and cosine, respectively, which the student should 
verify. 


1 ™. I 7 7 25 V R. (24 24 2 5) 9 n • ( 5 5 13V R • (12 12 1 3 

• a - \25>24>24)> P' V5"5) ~Ti ~T~ /• «*■ “ • VTTT) TT» T27 > P■ KT3) S~t S~J- 


5 . a: 

9. i 
27. a: 
33. a: 
37. a: 
39. a: 
41. a: 
51. (f 


(*• s' s)- * (r v I)' 3 * * * 7 - (# *■ ^)- (# *)■ 

11. 5. 13. f. 15. V-- 17. f. 19. V- 21. i. 23. J. 25. 

(H, |8: (A, A. tt)- 29. See Prob. 3. 31. See Prob. 1. 

(I, l }); |3: (i, i, f). 35. a: (*, |V3, §^3); 0: (*V3, V 3 , 2). 

(f, *V 2 i); 0 : (iV 21 , 5 V 2 I, f). 

(^V / 2, 1, v^); (3: (JV^, 1, v^2). 43. rf)- 47. (ff, 4^, 

(i, 1^3, |V3); 0: (1^3, V 3 , 2 ). 45. (A, A, «)• 49. (*V3/ V§, 2 ). 
V6, 2^6, 5). 53. (*V^5, i, iV65). 55. (§^3, *V2, |V6). 


Exercise 2. Page 8 

11. 10 ; 5 V 3 . 13. 6; 6 V 3 . 15. 2 k; kV 3 . 17. 2 h; 2 / 1 V 3 . 

19. 57°. 21. 28°. 23. 13°. 25. cos 70°. 27. cot 41°. 29. tan 52°. 

31. sin 49°. 33. esc 9°. 35. sec 23°. 37. cot 51°. 39. sin 87°. 

41. cos 11°. 43. cot 5°. 46. esc 9° 20'. 47. tan 12°. 49. sin 41°. 

51. tan 18° 5'. 53. .18; .98; .18. 55. .50; .86; .58., 57. .71; .71; 1.00. 


1. .2079. 
11. 1.052. 
21 . 22 ° 10 '. 
31. 64° 10'. 


Exercise 3. Page 10 

3. .6249. 6. 1.046. 7. .2334. 9. .0816. 

13. 2.583. 16. 31.26. 17. .0963. 19. 1.133. 

23. 31° O'. 25. 22° 10'. 27. 75° 30'. 29. 72° 20'. 

33. 38° 10'. 35. 69° 30'. 37. 32° 20'. 


1. .0825. 
13. .9856. 
23. .8189. 
33. 20° 16'. 
43. 50° 22'. 
53. 12° 50'. 


Exercise 4. Page 12 


3. .7298. 5. 

15. 1.097. 

25. 1.001. 

35. 48° 24'. 

45. 45° 43'. 
55. 12° 10'. 


.1957. 7. . 

17. .2259. 
27. 7.012. 
37. 52° 13'. 
47. 63° 45'. 
57. 60° 20'. 


9. .2487. 
19. .0207. 

29. 4° 45'. 

39. 45° 16'. 
49. 20° 57'. 
69. 32° O'. 


11. 1.426. 
21. 8.509.f 
31. 22° 24'. 
41. 32° 17'. 
51. 32° 34'. 
61. 52° 50'. 


* Answers to even numbered problems are furnished in a separate pamphlet when 
requested by the instructor. 

f If the agreement of Example 4, page 12, were followed, the result would be 8.510. 
However, we agree that when the tabular difference is very large, in interpolating we 
shall use the appr° x i m ate proportional parts listed in the column of proportional parts. 

1 


TRIGONOMETRY 


2 


Exercise 5. Page 14 


1 . 

.05553. 

3. 

8.4457. 

5. 

11. 

.10540. 

13. 

1.4988. 

15. 

21. 

32° 11'. 

23. 

1° 32'. 

25. 

31. 

7° 28'. 

33. 

11° 37'. 

35. 

41. 

73° 5'. 

43. 

CO 

O 

CO 

CO 

45. 

51. 

.57817. 

53. 

.52787. 

55. 

61. 

2.1446. 

63. 

12° 21.2'. 

65. 

71.* 

15° 28.5' 

to 15' 

3 30.5'. 

73. 


.64487. 

7. 

1.0497. 

9. 

.85491. 

.81378. 

17. 

.53807. 

19. 

1.2754. 

15° 28'. 

27. 

17° 43'. 

29. 

64° 16'. 

67° 40'. 

37. 

74° O'. 

39. 

17° O'. 

47° 4'. 

47. 

32° 6'. 

49. 

.68827. 

1.0359. 

57. 

.92315. 

59. 

.32240. 

20° 10.1'. 

67. 

12° 27.3'. 

69. 

61° 14.6' 

49° 23.7'. 

75. 

85° 29.5'. 

77. 

11° 8.7'. 


6. Page 16 


Note. Two legitimately different methods for the solution of a specified tri¬ 
angle may lead to slightly different results. The answers in this chapter are given 
as obtained by use of secants and cosecants, when applicable, for avoiding 
divisions in finding unknown sides. 


1. b = 79.00; c = 93.50; 57° 40'. 

5. a = 42° 40'; 0 = 47° 20'; 5.515. 

9. a = 26° 32'; 0 = 63° 28'; .5189. 

13. a = 54° 24'; 0 = 35° 36'; .5936. 

17. a = 1.577; b = .2723; 80° 12'. 

21. a = 61° 29'; 0 = 28° 31'; 2.618. 
25. b = .01675; c = .02120; 37° 49'. 
29. See Prob. 19. 

33. a = 13.089; b = 3.6943; 74° 14.3'. 
37. a = 61° 28.6'; 0 = 28° 31.4'; 2.6178. 


3. a = 12.16; b = 2.882; 76° 40'. 

7. b = 55.60; c = 63.91; 60° 28'. 

11. a = 61° 28'; 0 = 28° 32'; 2.618. 
15. b = .01676; c = .02120; 37° 49'. 
19. a = 58° 27'; 0 = 31° 33'; 73,290. 
23. See Prob. 13. 

27. See Prob. 17. 

31. b = 89.873; c = 96.446; 68° 43.3'. 
35. a = 26° 31.3'; 0 = 63° 28.7'; .51896. 
39. a = 54° 23.7'; 0 = 35° 36.3'; .59353. 


Exercise 7. Page 17 

Note. In this answer book, in any problem where the units (such as feet, 
pounds, etc.) in terms of which the results are specified are the same as the units 
for the data, no unit’s name, like foot or pound, will be attached to the result. 

1. 28.54. 3. 5092. 5. 13,880. 7. 1250. 9. 41.63; 65° 54'. 

11. 444.5. 13. 3852. 15. 40.88. 17. 16,579. 19. 7424. 


Exercise 8. Page 19 


1. sin a = Hi tana = -Vd sec a = 

3. sin a = 41; cot a = i; sec a = iv'il. 

5. esc ol — 4j2-; cos 0 = 7. cot 0 = cot a: 

9 . cos a = i; esc 0 = 8. 11 . esc a = i; cos 0 = f. 

19. esc 17°. 21. tan 38°. 23. cos 11°. 25. 53° 16'. 

29. a = 17.19; b = 18.15; 46° 33'. 31. 219.4. 

35. .91840. 37. 3.2571. 39. 9° 40.6'. 


17. sin 27°. 
27. 43.57. 
33. 58.5. 
41. .91486. 


* So far as Table XI indicates, sec a = 1.0377 if a is any angle between 15° 28.5' 
and 15° 30.5', inclusive. At least a siz-place table and a sfz-place value of sec a would 
be necessary to determine a to the nearest minute. 


ANSWERS 


3 


L S* 

15. logio 
23. logio 
31. logs 
39. 25. 

®1* T2T* 

67. §. 

79. 4. 


Exercise 9. Page 22 


3- r&. 

N - 4. 

W = .35. 
125 = 3. 
41. 216. 
53. 3. 
69. - 1 
81. 10. 


5 . 


loo- 


a 2. 


9. 5i li. ioi 13. log 3 N = 5. 


17. log6 iV = — 2. 19. log5 JV = 5. 

25. logio AT = - .4. 27. log 7 49 = 2. 

33. log 4 64 = 3. 35. log 2 * = - 5. 


43. 10,000. 
55. 5. 57. 3. 

71. - 4. 

83. 100,000. 


21. logio N = 
29. logs 64 = 
37. 100,000. 

45. 10. 47. b. 49. 1. 

59. §. 61. 4. 63. 3. 65. 4. 

73. 5. 75. 10. 77. 27. 

85. - 1. 87. 100. 89. i. 


1. .7781. 

11. 1.4065. 
19. - .4771. 


Exercise 10. Page 24 

1.4771. 5. .3680. 7. .5441. 9. 1.1549. 

13. .9542. 15. 1.4313. 17. .2817. 

21. .3820. 23. - .2720. 25. - .2288. 


Exercise 11. Page 26 


1. 

Char. = 

3; mant. = .95. 

3. Char. = — 2; 

mant. = .684. 

5. 

Char. = 

0; mant. = .587. 

7. 3. 9. 

- 3. 

11. 0. 

13. 

4. 


15. - 5. 

17. Char = 6; mant. = 

= 0. 

19. 

Char. = 

- 3; 

mant. = 0. 

21. Char. = 1; mant. = 

= 0. 

23. 

Char. = 

0; mant. = .3333. 

25. Char. = 1; mant. = 

= .67. 

27. 

1.6599. 


29. - 2 + .6599. 

31. 0. 33. 

1. 

35. - 1. 




Exercise 12. Page 28 



1. 

1.7259. 


3. 8.2122 - 10. 

5. 7.8609 - 10. 

7. 

0.9269. 

9. 

6.5539 - 

10. 

11. 3.7782. 

13. 5.6990 - 10. 

15. 

4.3345. 

17. 

6.0000 - 

10. 

19. 3.0414. 

21. 242. 

23. 

7.56. 

25. 

.838. 


27. .0000795. 

29. 1430. 

31. 

8,900,000. 

33. 

.000599. 


35. .00400. 

37. 2.29820. 

39. 

0.14270. 

41. 

8.04844 ■ 

- 10. 

43. 9.40875 - 10. 

45. 4.77525. 

47. 

6.25527 - 10. 

49. 

9.90363 ■ 

- 10. 

61. 3.61993. 

53. 6.00000. 

55. 

.69897. 

57. 

17.83. 


69. 48,460. 

61. .2674. 

63. 

.003097. 

65. 

4.577. 


67. 4159. 

69. .0003600. 

71. 

1.299. 




Exercise 13. 

Page 31 



1. 

3.4188. 


3. 0.8431. 

5. 9.0744 - 10. 

7. 

8.5872 - 10. 

9. 

5.1190. 


11. 2.7237. 

13. 8.4074 - 10. 

15. 

1.2713. 

17. 

2.0546. 


19. 9.9909 - 10. 

21. 4.6354. 

23. 

5.7853 - 10. 

25. 

5.6647. 


27. 1.7538 - 10. 

29. 1725. 

31. 

355,800. 

33. 

3.094. 


35. .0002162. 

37. .4693. 

39. 

846.4. 

41. 

37,870. 


43. 1.030. 

45. 9.738 • 10- 6 . 

47. 

1.568 • 10“ 13 . 

49. 

4.26865. 


51. 0.72605. 

53. 9.47898 - 10. 

55. 

0.67374. 

57. 

8.86666 - 

- 10. 

69. 9.87715 - 10. 

61. 1.78956. 

63. 

5.95274. 

65. 

9.79846 - 

- 10. 

67. 6.11414. 

69. 6.61877 - 10. 

71. 

9.69902. 

73. 

1645.5. 


75. 112.97. 

77. .013267. 

79. 

.60003. 

81. 

286,410. 


83. 1.9400. 

85. 2.1584 • 10- 7 . 

87. 

5.3911 • 10 -12 . 


tO Hco 


4 


TRIGONOMETRY 


Exercise 14. Page 33 

Note. Throughout this answer book, in problems where either four-place or 
five-place logarithms are suggested for use, the four-place answer is given in heavy 
black type. 

1 24.91; 24.909. 3. .07942; .079410. 5. .2009; .20086. 7. 61.10; 51.098. 
9. .8142; .81422. 11. .1406; .14061. 13. .003069; .0030681. 

16. .2486; .24851. 17. 1.047 • 10 4 ; 1.0464 • 10 4 . 19. .001172; .0011723. 

2l! !o01015; .0010149. 23. 280.7; 280.68. 26. 27.61; 27.609. 

27. - .007667; - .0076660. 29. - 2.627 • 10" 8 ; - 2.6266 • 10 8 . 

31. (a) 6.782 • 10 s ; 5.7834 • 10 8 : ( b ) 8.298; 8.2983. 

33. (a) 7.238 • 10“ 4 ; 7.2368 • 10~ 4 : (6) - .1520; - .15207. 

36. 1.5600 • 10 7 . 37. 5.6200 • 10~ 2 . 39. 6.3355. 41. 5.9523 - 20. 


Exercise 16. Page 35 


1. 6358; 5359.5. 

5. 66.19; 56.206. 

11. .9500; .94986. 

17. 28.93; 28.935. 

23. 41.46; 41.470. 

29. 37.21; 37.194. 

35. 4.771 • 10 4 ; 47,724. 
39. 5.176 • 10- 5 ; 5.1759 • 


3. 3.125 • 10- 5 ; 3.1256 • lO' 8 . 

7. 1.044; 1.0440. 9. 5.418; 5.4169. 

13. .4199; .41984. 15. .02154; .021544. 

19. - 4.476; - 4.4757. 21. 2.026; 2.0264. 

25. 16.72; 16.700. 27. 72.47; 72.455. 

31. .001352; .0013525. 33. .9388; .93896. 

37. - 1.916; - 1.9156. 
lO" 8 . 41. 1.041; 1.0412. 


43. .1266; .12658. 45. .5017; .50168. 

47. 5.558 • 10 -6 ; 5.5586 • 10~ 6 . 49. 93.84; 93.838. 

51 324.4; 324.29. 53. .4949; .49486. 55. $428.2. 67. $1408. 

69. 21.29; 21.288. 61. (a) .9066; .90630: ( b ) 1396; 1396.3. 

63. 236.1; 236.13. 65. - 940.2; - 939.88. 67. - .136; - .1366. 

69. 9.965; 9.9640. 71. 37.64; 37.631. 73. .2739; .27392. 75. .4194; .41943. 
77. 2.031 • 10 187 ; 2.0310 • 10 187 . 79. 3.617; 3.6173. 81. 145.6; 145.56. 


Exercise 16. Page 38 

1. 1.341; 1.3410. 3. 1.319; 1.3194. 5. - 5.195; - 5.1923. 

7. ±1.100; ±1.1001. 9.*18.1; 18.02. 

11. 2.617; - .617: 2.6166; - .6166. 13. 1.153 • 10 8 ; 1.1535 • 10 8 . 

15. 5.63; 5.634. 17. 29.5; 29.49. 

19. (a) 11.72; 11.722: ( b ) 613.9°; 613.91°. 

21. (a) .740; 740.7: ( b ) .936; .9357. 


Exercise 17. Page 40 

1. 4.317; 4.3176. 3. 1.291; 1.2911. 5. 2.303; 2.3026. 

7. - 1.449; - 1.4496. 9. - 14.2; - 14.20. 

11. 8.382; 1.474: 8.3822; 1.4743. 

13. Briggs, to natural, .4343; .43429: natural, to Briggs, 2.303; 2.3026. 

* The results are not reliable beyond the given digits unless Table XVI is used in 
the preliminary steps of the solution. 


ANSWERS 


5 


Exercise 19. Page 41 

3. 1.568. 5. - 1.503. 7. 1000. 9. .01. 11. 64. 13. ^. 15. 5. 17. 3. 

19. log 2 N = 5. 21. logic N = - 7. 23. 4. 26. - 1. 

27. Char. = 5; mant. = .68. 29. Char. = - 3; mant. - .965. 

31. Char. = - 1; mant. = .34. 33. 4. 35. - 2. 37. — 1. 

39. - 662.3; - 662.47. 41. .8628; .86272. 

43. .8222; .82196. 45. .01658; .016583. 


Exercise 20. Page 44 


1 . 

7. 

15. 

23. 

33. 

43. 

51. 

59. 

67. 

75. 


(a) .2136; (6) 
9.9191 - 10. 
9.1638 - 10. 


9.3296 - 10. 

9. 9.9404 - 10. 
17. 9.8472 - 10. 


26° 20'. 

57° 48'. 
9.82283 - 10. 
9.82678 - 10. 
0.10215. 

17° 4.8'. 

6 ° 1 . 8 '. 


25. 

35. 


47° 40'. 
84° 48'. 


27. 

37. 


45. 

53. 

61. 

69. 

77. 


9.71946 - 10. 
9.66350 - 10. 
9.98664 - 10. 
70° 24.3'. 

21° 35.5'. 


3. 9.8344 - 10. 

11. 9.9969 - 10. 

19. 9.3066 - 10. 21. 

21° 38'. 29. 39° 12'. 

64° 7'. 39. 5° 35'. 

47. 9.85088 - 10. 49. 
55. 9.73903 - 10. 57. 

63. 17° 21.0'. 65. 

71. 27° 1.4'. 73. 

79. 70° 5.8'. 81. 


6. 9.8022 - 10. 
13. 0.3911. 

9.9046 - 10. 
31. 53° 7'. 
41. 20° 38'. 
9.96817 - 10. 
9.99336 - 10. 
83° 23.0'. 

14° 24.1'. 

72° 24.8'. 


Exercise 21. 

1 . b = 21.34; c = 26.50; 53° 40'. 

5. a = 51.09; c = 97.20; 58° 18'. 

9. a = 4060; b = 1364; 18° 34'. 

13. a = 26° 41'; j8 = 63° 19'; 8.180. 

17. b = .05202; c = .05756; 64° 41'. 

21. b = 25.105; c = 34.170; 42° 43'. 

25. a = 18.138; b = 37.828; 64° 23'. 

29. a = 16.079; b = 34.273; 64° 52'. 

33. a = 26° 41.8'; /S = 63° 18.2'; 818.02, 
37. b = 1.6842; c = 1.9301; 60° 45.5'. 


Page 46 

3. a = 28° 30'; j8 = 61° 30'; .7523. 

7. a = 17° O'; j8 = 73° O'; .8732. 

11. b = 2.344; c = 2.738; 58° 53'. 

16. b = 48.76; c = 63.74; 40° 7'. 

19. a = 3° 15'; j8 = 86° 45'; 10.05. 

23. a = 43° 27'; 0 = 46° 33'; .31638. 
27. a = 42° 18.1'; /3 = 47° 41.9'; .85153, 
31. b = 2.6715; c = 3.1304; 58° 35.3', 
35. a = 1564.3; b = 1257.6; 51° 12.1', 
39. a = 1° 12.1'; <3 = 88° 47.9'; 103.42. 


Results in heavy type are those obtained by four-place computation. 

41. k = 512.2; 512.18: a = 55° 50'. 43. y = 83° O': k = 835.0; 834.90. 

45. a = 45° 55'; 45° 54.0': y = 88° 10'; 88° 12.0'. 

47. a = 65° 16': y = 92.50; 92.485: k = 77.40; 77.390. 

49. a = 402.5; 402.58: c = 539.3; 539.34: (3 = 29° 40'. 

51. a = .7348; .73492: b = 1.035; 1.0349: 13 = 74° 58'. 

53. a = 43.50; 43.507: 0 = 19° 54'; 19° 53.6': y = 131° 60'; 131° 50.8'. 

65. c = 8.60; 8.607: a = 44° 38'; 44° 37.8': y = 24° 32'; 24° 32.2'. 


Exercise 22. Page 48 

1. 159. 3. 7° 7'. 5. Pitch = .385; 37° 40'. 

9. Rad. = 34.0; side = 28.2. 11. Insc. = 195; circums. = 205. 

15. Insc. = 5.37; circums. = 10.7. 17. 2.814(10 4 ) sq. ft. 

21. 359.5. 23. 12,150 ft. 25. Height = 359; dis. = 324. 


7. 22.0. 
13. 99.2. 
19. 94.45 
27. 31.8. 


6 


TRIGONOMETRY 


1 . 


5. 


11 . 

15. 


19. 


Exercise 23. Page 52 


Ver. = 40.9; hor. = 131.3. 
29.6 ft.; 52° 58'. 7. 

17.9 IF; 15.5 N. 

.864 mi.; N 48° 34' W. 

144 mi.; S 40° 57' E. 


3. Ver. = .1238; hor. = .1072. 
.08457 ft.; 43° 25'. 9. 264 E; 280 N. 

13. 67.4 mi.; N 29° 2' E. 

17. 68.6 mi.; N 40° 27' E. 

21. 1209 ft.; S 61° 51' IF. 


Exercise 24. Page 54 

1. 128.4 up; 84.7 hor. 3. 22.07 E; 36.49 N. 5. 117.1 E; 295.9 S. 

7. 37.1 lb.; 28° 38'. 9. 171 lb.; N 18° 20' E. 11. 6741b.; S 18° 47' IF. 

13. 18.25 mi. per hr.; N 80° 32' E. 15. 741 lb.; N 60° 43' E. 

17. 486 1b.; S 48° 17' E. 19. 3571b.; N 22° 15' E. 21. 396 lb.; N 75° 46' E. 

23. 772 lb.; S 14° 3' E. 25. Ill lb. pressure; 100 lb. to drag. 

27. 1123. ' 29. (a) 202 1b.; (b) 208 lb. 31. 2181b. 33. 33.3 ft. 

Exercise 25. Page 58 

1. <. 3. <. 5. <. 7. >. 27. 5. 29. Vl3. 

31. 5. 33. 25. 35. V2- 37. 2. 43. - 3. 

Exercise 26. Page 60 

The answers for Problems 1 to 19 are not the only results which may be given. 

1. 405°; - 315°. 3. 480°; - 240°. 5. 300°; - 420°. 7. 90°; - 630°. 

9. 900°; - 180°. 11. 270°; - 810°. 13. 135°; - 225°. 

15. 360°; - 360°. 17. 135°; - 945°. 19. 330°; - 30°. 

Exercise 27. Page 63 

In Problems 1 to 17, only the sine, tangent, and secant are listed here, in this order. 
By taking reciprocals, check the other functions which you obtained. 


1. 

3.3.5 3 

5t 4,1 4' 

24. 24. 25 

2TT) ~7~t • 

C 5 • 5.13 

T3> T2) T 2 * 

7. 

3. _ 3. _ 5. Q 

5 1 4 t 4- S7 ‘ 

— Ht — > 2 r~- 

11- — it it — §■ 

13. 

|V2; 1; V2. 15. 

- ^V29; - f; |V29. 

17. - |V3; -V3; 2. 

Functions are given in the order sine, cosecant; tangent, cotangent; cosine, secant. 


19. 1; 1; none; 0; 0; none. 21. 0; none; 0; none; - 1; - 1. 

23. 1; 1; none; 0; 0; none. 25. 0; none; 0; none; 1; 1. 

27. — 1; — 1; none; 0; 0; none. 33. II or III. 35. II or IV. 

37. Ill or IV. 39. I or IV. 41. II. 43. II. 45. III. 47. II. 

Exercise 28. Page 65 

Only the sine, tangent, and secant are listed here, in this order; by taking recip¬ 
rocals, check the other functions which you obtained. 


3. 

_ 1. l\/q* — 

2 7 3 V °7 3 v °* 

5. 

tO|M 

1 

I— 1 

1 

7. 

- iV3; V3; - 2. 

9. 

- 1; - 1^; 1^3- 

11. 

- iV 2 ; 1; - V 2 . 

13. 

- h i^3; - fV3. 

15. 

1V3; - V3; -2. 

17. 

W 2; - 1; - ^2- 

19. 

- 1; - *V3; |V3. 

21. 

5 • 5.13 

T3? nr 7 T2- 

23. 

it ~ ~~ f- 

25. 

_ 4 • _ 4 • 5 

T7 373* 

27. 

- I; fV6; - 

29. 

- |V7; - *V7;t. 

31. 

- *V29; f; - 


ANSWERS 


7 


Exercise 29. Page 67 

1. 41°. 3. 52°. 6. 18°. 7. 47°. 9. 73°. 11. 63°. 13. 5°. 

In Problems 19 to 35, only the sine, tangent, and secant are listed, in this order; 
by taking their reciprocals, verify your other results. 

15. 80°. 17. 30°. 19. (|V3, - V§, - 2). 21. (- J, fV3, - |V3). 

23. (- |V2, 1, - V2). 25. (- |V3, - Vz, 2). 27. (f, |V3, fVjj). 

29. (- |V2, - 1 , V2). 31. (|V3, - V3, - 2). 33. (- 1 , - V2). 

35. (f, - fVjj, - |V3). 37. .88295. 39. - .60182. 41. .93252. 

43. - 1.4142. 45. - .88526. 47. - 3.6806. 49. .99039. 51. 1.7298. 

53. - .60529. 55. .92601. 57. .98315. 59. - .13687. 

61. - 3.872; - 3.8713. 63. -.08698; -.086978. 65. 21.53; 21.530. 

67. - .2866; - .28662. 69. - 1.224 • 10 4 ; - 12,237. 71. 61.33; 61.339. 

73. - 58. 75. 27. 77. 0. 79. 0. 81. .0013. 

Exercise 30. Page 68 

1. - sin 35°. 3. - tan 12°. 5. cos 128°. 7. sec 35°. 

9. - esc 148°. 11. - tan 130°. 13. cos B. 15. - sin W. 


Exercise 31. Page 71 

to 29, only the cosine, cotangent, and cosecant are listed, in this 
check the other results. 


In Problems 1 

order. By inference from these, 

1. sin 42°; tan 42°; sec 42°. 

5. - cos 13°; - cot 13°; esc 13°. 

9. sin 12°; - tan 12°; - sec 12°. 

13. — cos 15°; cot 15°; — esc 15°. 

17. cos 37°; - cot 37°; - esc 37°. 

21. — cos 44°; cot 44°; — esc 44°. 

25. - sin 23°; - tan 23°; sec 23°. 

29. — cos 2°; — cot 2°; esc 2°. 37. 

43. — tan 9. 45. cot d. 47. 

53. tan 9. 55. sin 6. 


3. - sin 37°; - tan 37°; sec 37°. 

7. — cos 41°; cot 41°; — esc 41°. 

11. cos 36°; - cot 36°; - esc 36°. 

15. sin 35°; - tan 35°; - sec 35°. 

19. sin 43°; - tan 43°; - sec 43°. 

23. - sin 23°; - tan 23°; sec 23°. 

27. cos 35°; - cot 35°; - esc 35°. 

tan 9. 39. — cos 9. 41. — esc 9. 

— esc 9. 49. tan 9. 51. sin 9. 

57. sec 9. 59. — cos 9- 


Exercise 32. Page 72 

In problems where the values of functions are asked for, they are listed in the order 
sine, cosecant, tangent, cotangent, cosine, secant. 


1 . 

11 . 

15. 

21 . 

27. 

35. 

37. 

41. 

49. 


10; 5. 9- (~~ ii> tti V) 

(- iV2, - V2, - 1, - 1, i^2, V2). 

(A, 2, - iVs, - V3, - aV3, - |V3). 

(— A, — ~~ T2> ~~ xfi if)- 

— |V3. 29. - i 31. §V3. 

(- a, - 2, |V3, V3, - 1^3, - fV3). 
(- a, - 2, - iV3, - V3, aV 3, fV3). 

— .22495. 43. — 1.1504. 45. 

— .7275; - .72732. 51. - cos 9. 


5 5 13 ^ 

TT2) T3A 5 /• 

13. (0; none; 0; none; 1; 1). 


17. III. 

23. - Y/% 
33. - iVS- 


- 1.4554. 
53. tan 9. 


19. 

25. 


III. 

\V3. 


47. - .03490. 
65. — tan 9. 


8 


TRIGONOMETRY 


1. |tt. 

15. - |x. 
29. 132°. 
46. A?r. 
55. 0. 

65. 2.160. 
75. cos 0. 


Exercise 33. Page 74 


3. r. 

17. - 2t r. 
31. - 180°. 
47. fx. 


5. |X. 7. £X. 

19. 30°. 21. 45°. 

33. 114° 35V 35. 
49. |V3. 


57. 1. 59. 1. 

67. 3.199. 69. 4.970. 

77. cot 0. 79. tan 0. 


9. fx. 11. fx. 13. itt. 

23. 90°. 25. 135°. 27. 120°. 

143° 14'. 37. - 140°. 39. 15°. 


51. - 2. 63. - 1. 

61. §V3. 63. - 1^3- 

71. 5.501. 73. - cos 0. 

81. cos 0. 83. esc 0. 


Exercise 34. Page 76 

1. 23.0 ft. 3. 5.71 ft. 6. 34.1 in. 7. 2.41 rad. 9. 640 ft. 

11. 3.65 rad. 13. 179 in. 15. 37.7 in. 17. 113 ft. 19. 15 ft. 

21. 19.2 in. 23. 36.8 in. 26. 642 yd. 27. 14.4 in. 

29. co = .803 rad. per sec.; v = 27.3 ft. per sec. 31. (a) .105; (6) .00873. 
33. .291 mi. 35. (a) 1800; ( b ) 286. 


Exercise 36. 

7. - 360°; - 180°; 0°; 180°; 360°. 

11. - 360°; - 180°; 0°; 180°; 360°. 

15. - 90°; 270°. 

19. - 270°; - 90°; 90°; 270°. 


Page 84 

9. - 360°; 0°; 360°. 

13. - 180°; 180°. 

17. - 360°; 0°; 360°. 

21. - 360°; - 180°; 0°; 180°; 360°. 


Exercise 38. Page 88 

1. *x. 3. fx. 5. fx. 7. 1.15t r. 9. 450°. 11. 57.30°. 

13. - 22.92°. 15. ^x; ^x; Ifx. 17. 1.875. 19. IV 3 . 21. - V 3 . 


In Problems 23 to 27, only the sine, tangent, and secant are listed, in this order. 


23. 

- 

V3, - 2). 

25. (0, 0, - 1). 

27. (- i iV3, - |V3). 

33. 

— sin 0. 

37. 

.83646. 41. .63116. 

46. - .50622. 

35. 

sec 0. 

39. 

.85934. 43. .09266. 

47. - .89605. 




Exercise 39. Page 90 


7. 

tan 0. 

9. 1. 

11. sin 0. 13. sin 2 0. 

15. 1. 17. cos 0. 


In Problems 19 to 35, only the cosine, cotangent, and cosecant are listed, in this 
order; check your other results from these. 


19. (if, 4). 

25. 

31. (i lV2, fV2). 

37. - 18. 


21. (- H, ¥> - ¥-)• 23. Of, if). 

27. (- f, - f, *). 29. (- M, - ^)- 

33. (- iV 5 , 1, - 1V5). 35. (|V6, - 2V6, - 5 ). 


39. - if. 


41. sin x 


± 


tan x 

Vl -f tan 12 x 


43. sin x = ± Vl — cos 2 x', tan x = zb 


vT 


cos 2 x 


sec x = 


cos x 


45. sin x 


v sec 2 x — 1 


tan x = zb Vsec 2 x — 1- 


sec x 











ANSWERS 


9 


47. 

55. 


1 

sin x 
sin x. 


49. cos 2 x. 


51. 


1 


sm x cos x 
67. sin x cos x. 59. sin 2 x. 61. 


53. cos x sin x. 


1 


sin x cos x 


63. 


COS X 

sin x 


Exercise 41. Page 96 


1. 

45°; 

; 225°. 

3. 

0°. 

6. 

o 

O 

r—H 

330°. 

i 

7. 60° 

; 240°. 9. 225°; 315°, 

11. 

45°; 

; 315°. 


13. 

none. 


15. 

none. 

© 

t-H 

O 

t-H 

iO 

t-H 

b 

*o 

o 

00 

CN 

rH 

19. 

58° 

20'; 238° 20'. 




21. 

145 

° 20'; 

214° 40'. 

23. 

30°; 

; 150° 

; 210°; 

330° 



25. 

45°: 

; 135° 

; 225°; 315°. 

27. 

60°; 

; 120° 

; 240°; 

300° 



29. 

45° 

; 135° 

; 225°; 315°. 

31. 

60°; 

; 135° 

; 240°; 

315° 



33. 

60° 

; 180° 

; 300°. 

35. 

45°: 

; 120° 

; 225°; 

300° 



37. 

30° 

; 150° 


39. 

45°; 

; 135° 

; 225°; 

315° 



41. 

60° 

; 180° 

; 300°. 

43. 

0°; 

90°; 180°. 




45. 

0°; 

180°; 

225°; 315°. 

47. 

None. 





49. 

30°; 

; 90°; 

150°; 210°; 270°; 330°. 

61. 

45°: 

1 90°; 

135°; 225°; 

270°; 

315°. 

63. 

0°; 

45°; 135°; 180°; 225°; 315°. 

55. 

30° 

! 90°; 

150°; 210°; 

270°; 

330°. 

57. 

196 

° 19'; 

343° 41'. 

69. 

100 

° 14'; 

280° 14 

'; 28 

° 26'; 

208° 

26'. 


61. 

None. 

63. 

207 

° 56'; 

332° 4'. 




65. 

0°; 

48° 11'; 311° 49'. 


Exercise 42. Page 98 

1. 90°; 210°; 330°. 3. 270°. 5. None. 7. 0°; 45°; 180°; 225°. 

9. 270°. 11. 63° 26'; 116° 34'; 243° 26'; 296° 34'. 

13. 60°; 120°; 240°; 300°. 17. 60°; 180°; 300°. 21. 0°; 180°. 25. 0°; 270°. 

15. 30°; 210°. 19. 210°; 270°; 330°. 23. 60°; 300°. 27. 0°; 90°. 

29. 180°. 31. 30°; 150°. 33. 60°; 120°; 240°; 300°. 35. 45°; 135°; 225°; 315°. 
37. 60°; 120°; 240°; 300°. 39. 233° 8' (or 233° 7', depending on method). 


Exercise 43. Page 99 


In Problems 5 to 9, the cosine, cotangent, and cosecant are listed, in this order. 


6 ( 7 7 25\ 

• V 2~5> 24 J- 

Vl -f- tan 2 9 
tan d 

21. 196° 20'; 343° 40'. 
27. 30°; 90°; 150°; 270° 
33. 26° 34'; 206° 34'. 

65. 30°; 60°; 150°; 210 


7 (— 2S — T 25 ~\ 

V ~T~ J 2 4) 24/* 

13. 30°; 150°. 

15. 135°; 225°. 

23. 194° 35'; 345° 25'. 
29. 180°. 


9. (- iV3, - V3, 2). 
17. 120°; 300°. 

19. 33° 30'; 146° 30'. 

25. 60°; 300°. 

31. 0°; 180°; 210°; 330°. 

69. 210°; 330° 
180°; 225°; 315° 


35. None. 67. 30°; 45°; 210°; 225°. 
°; 300°; 330°. 67. 0°; 45°; 135° 


Exercise 44. Page 105 

The functions are given in order, sine, cosine, tangent, and cotangent, in any prob¬ 
lem where their values are requested. 

9. |(V2 + v'6); J(V6 - V2); (2 + V§); (2 - v"3). 

11 . - V 2 ); - i(V 2 + V6); (V§ - 2); - (2 + Vs). 

13. - i(V2 + V6); i(V6 - y/2); - (2 + V3); (V3 - 2). 








TRIGONOMETRY 


10 


15. f (cos a + sin a). 19. f V2(cos A + sin A). 

17 1 + tan gj V3 cot a + 3 

1 — tan 0 V 3 — 3 cot a 


23. — cos a. 
25. — tan a. 


29. ( 0 ) (1 + V3); (b) - (2+ V3). 31. - 1. 33. cot (- 85°). 

35 . l. 37. cos (A - 40°). 39. sin (A - 50°). 41. cos 2 B. 

55. (ex + (fff> — wr> — ^r)j ( a fi)' ( — ttt)' 

67. (a + 0): (**, ft); (« — /3): (ff, ft, «). 

59. (ex + 0): (fff, ^rr)> (& ~ $)• (~ Mr)- 


61. I. 63. IV. 


3 tan A — tan B 
3 tan A tan B + 1 


Exercise 45. Page 108 

In problems where the values of functions are asked for, they are listed in the order 
sine, cosine, tangent, and cotangent. 

3 . (- iVS, \, - V3). 5 . (- l - V3). 9 . (*V 5 , - |V2, - 1). 

11. (-l'V3, - *V2,1). 17. (*V2,*V2,f). 19. (- *VlO, *^10, - *). 

21. sin 70°. 23. cos 160°. 25. 2 cos 2 20°. 27. 1 + cos 30°. 

29. 2 sin 2 %B. 31. cos 2H. 33. tan 80°. 35. sin 70°. 37. tan 2 f0. 

2 tan 3a cot 2 3a — 1 


63. 2 sin 3a cos 3a; 2 cos 2 3a — 1; 


1 — tan 2 3a’ 2 cot 3a 


65. ± Vf(l — cos 8x); =b Vf(l -f cos8x); ± y^- 

2 tan 5x 


— cos 8x 


± 


67. 2 sin 5x cos 5x; (2 cos 2 5x — 1); 


+ cos 8x’ 
cot 2 5x — 1 


VS 


+ cos 8x 


cos 8x 


1 — tan- 5x’ 2 cot 5x 


69. =fc v^^(l — cos 6A); ± Vj(l -|- cos 6A); ± yj- 


— cos6A 


+ cos 6A ’ 


vs 


cos 6A 


cos 6A 


_ . 3a 3a /, 0 . , 3a\ 2 tan fa cot 2 fa — 1 

71. 2 sm — cos -?r; (1 — 2 sin 2 ); ---—; — - - - 

2 2 ’ \ 2 /’ 1 — tan 2 fa’ 2 cot fa 


73. 4 cos 3 x — 3 cos x. 75. 


3 tan x — tan 3 x 
1 — 3 tan 2 x 


77. 1 — 8 sin 2 x + 8 sin 4 x. 


11 . cos 20 — cos 80. 

17. cos 60 — cos 80. 

23. — 2 sin 55° sin 25°. 
29. 2 sin 100° cos 40°. 
35. 2 sin 4x cos x. 

41. 2 cos cos fy. 


Exercise 46. Page 111 

13. cos 120 + cos 20. 

19. cos 50 + cos 0. 

25. 2 cos 140° cos 10°. 

31. 2 cos 3x cos x. 

37. — 2 cos 2x sin x. 
i. — V8. 57. cot fB t£ 


15. sin 50 — sin 0. 

21. cos 80 — cos 100. 

27. - 2 cos 70° sin 40°. 
33. — 2 sin 2x sin x. 

39. — 2 sin fy sin fy. 
l iB. 69. cot i(0 - a). 


Exercise 48. Page 116 

1. 0°; 90°; 180°; 270°. 3. 90°; 270°. 5. 90°; 210°; 330°. 

7. 20°; 40°; 140°; 160°; 260°; 280°. 11. 15°; 75°; 135°; 195°; 255°; 315°. 

9. 37f°; 82f°; 127§°; 172f°; 217§°; 262f°; 307§°; 352|°. 13. 0°; 180°. 

15. 240°. 17. 90°. 19. 180°. 21. 30°; 90°; 150°; 270°. 

23. 60°; 180°; 300°. 25. 60°; 120°; 240°; 300°. 27. 60°; 90°; 270°; 300°. 


























ANSWERS 


11 


29. 30°; 90°; 150°. 31. 0°; 60°; 300°. 33. 90°; 270°. 35. 0°; 45°; 180°; 225°. 
37. 45°; 135°; 225°; 315°. 39. 15°; 75°; 105°; 165°; 195°; 255°; 285°; 345°. 

41. None. 43. 22|°; 67i°; 112|°; 157i°; 202§°; 247i°; 292^°; 337§°. 

45. 0°; 90°; 180°; 270°. 47. 70°; 110°; 190°; 230°; 310°; 350°. 49. 135°; 315°. 

51. 0°; 120°. 53. 60°; 300°. 55. 60°; 300°. 67. 0°; 90°; 180°; 270°. 

69. 36°; 108°; 180°; 252°; 324°. 61. 0°; 60°; 90°; 120°; 180°; 240°; 300°. 

63. 0°; 22J°; 67i°; 112*°; 157|°; 180°; 2021°; 247i°; 292|°; 337|°. 

65. 0°; 60°; 120°; 135°; 180°; 240°; 300°; 315°. 

67. 15°; 45°; 135°; 165°; 255°; 285°. 69. 120°. 

71. 20°; 40°; 80°; 100°; 140°; 160°; 200°; 220°; 260°; 280°; 320°; 340°. 

73. 26° 34'; 153° 26'; 206° 34'; 333° 26'. 

75. 31° 43'; 35° 47'; 121° 43'; 125° 47'; 211° 43'; 215° 47'; 301° 43'; 305° 47'. 
77. 15°; 45°; 75°; 195°; 225°; 255°. 79. 30°; 150°; 210°; 330°. 

81. 45°; 90°; 135°; 225°; 270°; 315°. 83. 75°; 165°; 255°; 345°. 

85. 22^°; 30°; 67|°; 90°; 1121°; 150°; 157J°; 202§°; 247§°; 270°; 292§°; 337§°. 
87. 0°; 30°; 90°; 150°; 180°; 210°; 270°; 330°. 


Exercise 49. Page 118 

When the values of the functions of an angle are given, they are listed in the order 
sine, cosine, tangent, and cotangent. 

1 . i(V2 + V 6 ); 1 (V 6 - V 2 ); (2 + V3); (2 - Vjj). 

1 V 2 - V3; iV 2 +V 3 ; V 7 - 4 V 3 ; V 7 + 4 V 3 . 


3. 

7. 

13. 

17. 

23. 

49. 

55. 

61. 

65. 

67. 

71. 

77. 


(A 

— 2 sin 50° sin 25°. 

\ cos 6/3 + \ cos 4/3. 
60°: 150°; 240°; 330 


(251 

); i/3: (iV5, |V5, i). 


(rff^lO) 3^'V / 10 ) ^). 


*)• 

15. 2 cos f a cos \a. 

19. sin 10° — sin 2°. 21. sin 100 + sin 8t 

25. 2 sin i(3 a + 2/3) cos i(3o; — 2/3). 

53. 0°; 90°; 180°; 


51. 60°: 180°; 300°. 


90°; 270°. 57. 45°; 135°; 225°; 315°. 

35° 16'; 144° 44'; 215° 16'; 324° 44'. 

0°; 22|°; 67i°; 90°; 1121°; 157i°; 180°; 202|°; 

45°; 135°; 225°; 315°. 69. 60°; 300°. 

0°; 60°; 120°; 180°; 240°; 300°. 73. None, 

tan a — tan /3 — tan 7 — tan a tan /3 tan 7 
1 + tan a tan (3 + tan a tan 7 — tan /3 tan 7 


270 c 
300 c 


59. 0°; 60°; 180 . 

63. 0°; 120°; 240°. 
2471°; 270°; 292i°; 337 


75. 30°; 210°. 
79. 3 sin /3 — 4 sin 3 /3. 


Exercise 50. Page 122 


1. 165° 50'. 3. 97° 50'. 5. 2.646. 

11. 10.54. 13. 41° 25'. 15. 57° 7'. 

21 . a = 55° 46'; /3 = 82° 49'; 7 = 41° 25'. 

23. a = 21 ° 47'; /3 = 120°; 7 = 38° 13'. 

25. a = 34° 1'; /3 = 37° 59'; 7 = 108° O'. 


7. 7.071. 9. 8.062. 

17. 120°. 19. 117° 49'. 

27. 3.83. 29. 9.94. 31. 9.88. 

35. 13.3; 4.8. 37. 115 mi. 

39. 394. 41. 58.7. 


Exercise 51. Page 126 

1. a = 5.578; c = 4.084; 45°. 3. a = 26.51; h = 18.65; 73° 10'. 

6. a = 5.577; c = 7.321; 78° 20'. 7. a = 10.03; c = 17.40; 104°. 

9. b = 302.4; c = 137.9; 126° O'. 







12 


TRIGONOMETRY 


ll.*a = 13.10; 13.100: 6 = 19.35; 19.345: 84° O'. 

13. b = .4290; .42897: c = .8744; .87436: 102° 50'. 

15. b = 7.518; 7.5175: c = 11.81; 11.814: 132° 11'. 

17. a = .1852; .18517: c = .4566; .45548: 70° 50'. 

19. b = .07758; .077598: c = .06650; .066503: 58° O'. 21. 2.80. 

23. b = 35.408; c = 34.700; 76° 10'. 25. b = 242.39; c = 339.55; 123° O'. 

27. a = .0037760; b = .011107; 80° 11'. 29. a = b sm n a . 31. b = — sm 

sin p sin a 


Exercise 52.* Page 131 

1. a = 23° 35'; 7 = 126° 25'; 8.048. 

3. a = 37° 20'; |8 = 77° 10'; 606.5: or, a = 11° 40'; /3 = 102° 50'; 202.2. 

5. a = 30°; /3 = 75°; 3.622. 

7. No solution. 9. /3 = 14° 29'; 7=7° 51'; 13.66. 

11. a = 21° 4'; 7 = 44° 26'; 5.136. 13. No solution. 

15. a = 107° 40'; 107° 39.5': 7 = 29° 50'; 29° 50.5': 23.65; 23.555. 

17. a = 64° 48'; 64° 46.9': (3 = 61° 52'; 61°53.1': 271.9; 271.82: or, 

a = 8° 32'; 8 ° 33.1': j 8 = 118° 8'; 118° 6.9': 44.59; 44.678. 

19. No solution. 

21. a = 81° 15'; 81° 15.6': 7 = 61° 62'; 61° 51.4': .08442; .084415: or, 

a = 24° 59'; 24° 58.4': 7 = 118° 8'; 118° 8 . 6 ': .03607; .036058. 

23. j 8 = 31° 21'; 31° 19.7': 7 = 35° 19'; 35° 20.3': 6.551; 6.5460. 

25. a = 47° 26': 7 = 85° 8 ': 1834; 1834.5. 27. No solution. 

29. j 8 = 33° 23'; 33° 23.0': 7 = 120° 40'; 120° 40.0': 2.512; 2.5121: or, 

& = 146° 37'; 146° 37.0': 7 = 7° 26'; 7° 26.0': .3778; .37784. 

31. /3 = 30° 43.9'; 7 = 125° 8.7'; 2.5399: or, (3 = 149° 16.1'; 7 = 6°36.5'; .35745. 
33. j 8 = 109° 28.6'; 7 = 29° 22.2'; 23,826. 

35. a = 67° 32.7'; 7 = 60° 8.2'; 2.8046: or, a = 7° 49.1'; 7 = 119° 51.8'; .41281. 
37. /3 = 19° 40.4'; 7 = 124° 51.6'; 13.925. 39. 57° 54'. 


Exercise 53.* Page 135 

1 . j8 = 91° 26'; 91° 25.9': 7 = 20° 34'; 20 ° 34.1': 358.9; 358.94. 

3. a = 10° 52'; 10° 52.0': |8 = 126° 8'; 126° 8.0': 369.1; 369.01. 

5. |8 = 50° 8'; 50° 7.4': 7 = 21° 52'; 21° 52.6': .8142; .81424. 

7. |3 = 34° 8'; 34° 7.7': 7 = 76° 12'; 76° 12.3': 22.00; 21.995. 

9. a = 31° 39'; 31° 39.7': 7 = 104° 53'; 104° 52.3': .009906; .0099080. 

11. a = 12° 9'; 12° 9.1': 7 = 144° 35'; 144° 33.9': 1.758; 1.7576. 

13. a = 134° 42.2'; (3 = 25° 57.2'; 10.004. 

15. a = 48° 40.6'; 7 = 73° 43.8'; 1.1053. 

17. a = 27° 59.8'; 7 = 93° 52.8'; 56122. 

19. a = 111 ° 44'; 0 = 38° 16'; 6.456. 

21. a = 27° 31'; 0 = 112° 29'; 13.92. 23. a = 48° 8 '; 7 = 21° 52'; 63.10. 

25. b = 103.5; 103.49: c = 247.8; 247.86: 7 = 135° 43'. 

27. a = .1760; .17595: c = .1897; .18970: j 8 = 74° 22'. 

29. a = 35° 24'; 35° 23.8': /3 = 76° 7'; 76° 7.2': 723.3; 723.37. 31. 423. 

* Answers obtained by four-place computation are in heavy type. 





ANSWERS 


13 


Exercise 54. Page 139 

I. a = 27° 40'; 27° 39.6': 0 = 45° 14'; 45° 14.2': 7 = 107° 6'; 107° 6.2'. 

3. a = 46° 30'; 46° 30.2': 0 = 39° 24'; 39° 24.0': 7 = 94° 6'; 94° 5.8'. 

5. a = 15° 44'; 15° 44.0': 0 = 13° 4'; 13° 3.0': 7 = 151° 12'; 151° 12.8'. 

7. a = 31° 26'; 31° 25.8': 0 = 65° 32'; 65° 32.4': 7 = 83° 2'; 83° 1.6'. 

9. a = 27° 24'; 27° 24.4': 0 = 30° 34'; 30° 35.0': 7 = 122° 2'; 122° 0.6'. 

11. a = 45° 16'; 45° 17.6': 0 = 73° 8'; 73° 8.4': .7 = 61° 34'; 61° 34.0'. 

13. a = 43° 6'; 43° 6 . 6 ': 0 = 31° 34'; 31° 35.2': 7 = 105° 18'; 105° 18.0'. 

15. at = 41° 44'; 41° 44.0': 0 = 23° 4'; 23° 4.6': 7 = 115° 12'; 115° 11.6'. 

17. a = 53° 2'; 53° 1.8': 0 = 86° 6'; 86 ° 6.0': 7 = 40° 62'; 40° 52.2'. 

19. a = 27° 28.2'; 0 = 79° 57.2'; 7 = 72° 34.4'. 25. 99° 36'; 99° 35.6'. 

21. a = 72° 23.4'; 0 = 56° 54.4'; 7 = 50° 42.0'. 27. 58° 20'; 58° 19.4'. 

23. a = 96° 40.0'; 0 = 38° 4.6'; 7 = 45° 15.6'. 29. 65° 30'; 65° 30.4'. 

31. See Prob. 19. 33. 84° 24'. 

Exercise 55. Page 141 

1. 20. 3. 2.25. 5. 8.817. 7. 25. 9. 41.57. 11. 89.44. 

13. 7.552; 7.5502. 15. .1108; .11078. 17. 274.9; 274.91. 

19. .01467; .014666. 21. .3935; .39355. 23. 12.57. 25. 4892; 4891.0. 

Exercise 56. Page 142 

1. 39.94. 3. 38° 12'. 6. 125° 6 '. 7. b = 38.07; c = 26.82; 107° 53'. 9. 32° 6 '. 

II. « = 36° 52'; 7 = 120° 48'; 85.90: or, a: = 143° 8 '; 7 = 14° 32'; 25.09. 

13. a = 6 ° 4'; 7 = 9° 36'; 31.71. 15. 0 = 66 ° 25'; 7 = 23° 35'; 45.82. 

17. cc = 90°; 7 = 70°; 1879. 

19. b = 27.77; 27.771: c = 29.25; 29.243: 70° 29'. 

21. a = 27° 52'; 27° 51.9': 0 = 71° 48'; 71° 49.1': 2.774; 2.7734. 

23. a = 64° 55'; 64° 55.3': 0 = 67° 44'; 67° 43.7': 164.2; 164.19: or, 

a = 116° 5'; 115° 4.7': 0 = 17° 34'; 17° 34.3': 53.54; 53.565. 

25. None. 27. None. 

29. a = 32° 34'; 32° 33.2': 0 = 104° 12'; 104° 11.8': 7 = 43° 14'; 43° 15.2'. 

31. a = 32° 45'; 32° 45.1': 7 = 57° 16'; 57° 14.9': 3.016; 3.0165. 

33. 27° 17'; 152° 43'. 35. 184. 37. 6.128. 39. 1513. 41. 73° 40'. 

43. 9812. 45. 4880. 47. 81.4. 49. 2833 ft. 51. N 83° 33' W. 53. 659. 

56. 5553. 57. 174.3. 59. 800.9. 61. 294. 63. 26.7 lb.; S 83° 16' E. 

65. 872 1b.; S 59° 56' E. 67. 600 1b.; 0° 51'north of vertical. 

69. Second force: S 3° 50' W; resultant: S 33° 45' W. 

71. Mag. 708; S 86 ° 42' E. 73. 4.3 mi. per hr.; 63° 0'from downstream. 

75. 81° 50' from downstream. 

77. 48° 19'; 77° 20'; 54° 21'; 48.53, opposite 48° 19'. 

79. 187.8 ft. 81. 68.0; 17.3; 60.5. 83. 26.5; 44.1; 53.0. 

93. Distance: h cos 2 a cot or, equation: k = h cos 2 a. 95. 395 ft. and 35 ft. 

Exercise 57. Page 150 

The polar coordinates given in any answer are not the only correct coordinates which 
might be given. 

1. (3, 540°); (- 3, 0). 3. (5, 360°); (5, 0). 5. (1, £tt); (0, - 1). 


14 


TRIGONOMETRY 


7. (3, Itt); (fV2, *V2). 9. (^2, 315°); 


13. (V2, 315°); (1, - 1). 15. 

17. (2, 50°); (1.286, 1.532). 19. 

21. (V2, 45°); (- V2, 225°). 23. 

25. (2, 90°); (- 2, 270°). 27. 

29. (2, 240°); (- 2, 60°). 31. 

33. (5°, 36° 52'); (-5, 216° 520- 35. 

37. (5°, 233° 80; (- 5, 53° 80- 39. 


(1, - 1). 11. (3, 270°); (0, - 3). 
(3, 432°); (.927, 2.853). 

(6, 465°); (- 1.553, 5.795). 

(3,0°); (- 3, 180°). 

(V2, 225°); (- V2, 45°). 

(2, 270°); (- 2, 90°). 

(17, 61° 550; (~ 17, 241° 550- 
(25, 286° 160; (- 25, 106° 160- 


Exercise 59. Page 153 

1. y = arctan f. 3. y = arccsc (—5). 5. 6 = arcsin f. 7. 6 = arccot x. 

The answers given for Problems 9 to 41 are not the only correct results. 


9. 

0; 2t r; - 

• 2tt; - 

47T. 



11. 


|7r; - 

- Itt; 

- \TT. 



13. 

i*r; 


- §tt; - 

- fTT. 



15. 

Itt; 

Itt; ~ 

- stt; 

- It r. 



17. 

i7r; 

|tt; 

- Itt; - 

- -V"7r. 



19. 

0; 7T 

; - tt; - 

27T. 



21. 

Itt; 


- 

— TpTT • 



23. 

57r; 

!-7t; - 

- 

- fTT. 



25. 

Itt; 

ht-tt; 

- iTr; 

- frr. 



27. 

Itt; 

|x; - 

- Itt; 

- |7T. 



29. 

|tt; 

|tt; - 

- - 

JTT. 31. 

|tt; 

fTT 

) 

4-tt; - 

- |7T. 

33. 

No value. 


35. 

No value. 


37. 

14° 

10' 

; 345° 50 

r 

39. 

32° 40'; 

212° 

40'. 

41. 

236° 

1 50'; 

303° 10'. 

43. *. 

45. 

1 

5 • 

47. f. 

49. 

± f. 

51. f. 

53. 

1 7 
15* 

55. 

25 


67.1. 

59. 

1 


61. 


X 


63. 

X 


7 


X 


X 



Vl 

— X 2 


VI + 

X 2 

65. 

Vi 

+ X 2 - 





67. 

x = 

arcsin 

b or 

arcsin (- 

- i). 


69. 

X = 

arctan f, or arctan 5. 



71. 

X = 

arccos 

1 

4* 





Exercise 60. Page 156 


1. 

b r. 

1=: 

CO 


5. 

b r. 

7. 

iTT. 

9. 0. 


ii. 

0 . 

13. 

- §7T. 

15. — l-n 


17. 

fTT. 

19. 

TT. 

21. - 

ix. 

23. 

Jtt. 

25. 

- §7T. 

27. - TT. 


29. 

15° 10'. 

31. 

b 

o 

t-H 

33. - 

17° 

3'. 35. 

1 

3 • 

37. 

W3- 

39. 

V 3 . 

4i: 

fvOs. 

43. 

|V3. 


45. - 

6. 

47. 

- ^V2. 

49. 

0 . 


51. 

JV3. 

53. 



55. - 

1 

5* 

57. 

1 

3* 

59. 

3 

5* 


61. 

24 

2 5* 

63. 

3 

4* 


65. - 

119 
16 9* 

67. 

iVl 

69. 

1 

3* 


71. 

68°. 

73. 

110°. 


75. - 

O 

o 

(M 

77. 

xV 1 — y 2 

- yVl - 

x : 

2 . 79. 

y - x 

. 81. ? + » • 

83. 

2 y 

— X 




xy + 1 


t - xy 


1 + 

■ 2zy 


AH 

0^2 1 

A7 

1 

- X 2 

ao i/i 

ai 

v / 1 

- V\ - 

- X 2 





2x 


2 V A I 


V 

2 


93. 

xy V 1 — 

z 2 + vr- 


x 2 Vl 

- y 2 Vl 

- z 2 

+ yzV 1 - 

x 2 — xzV 1 

- I/ 2 . 


95. 

3x — 4x 3 

. 


97. 

2a. 


99. (a - 

-0). 




Exercise 62. Page 159 

17. *V2. 19. i 21. \V%. 23. *V2. 

25. 1. 27. None. 29. ^^21. 31. 5. 33. 0. 






















ANSWERS 


15 


1 . 

13. 

25. 

35. 


1 . 

11 . 

15. 

19. 

23. 

27. 


29. 


1 . 

7. 


11 . 

13. 

15. 

17. 

19. 

23. 


25. 

29. 

31. 

33. 

35. 


1 . 

5. 

11 . 


Exercise 63. Page 161 

- 1. 3. - 1. 5. 0 7. ± 7 i. 9. ± 5 i. 11. ± 5iV2. 

1 + 80 15. 14. 17. - 4. 19. 30. 21.- 11. 23. 9 + ;V§. 

- -is - MO 27. 1 + 2 i. 29. * - 31. i. 33. - 60 

x = Sj y — 2. 37. x = — 2; y = 3. 39. — -^fO 41. — gO 


Exercise 65. Page 164 


f + -|iV3. 3. 4. 5. 

6\/2(cos 45° + i sin 45°). 

3 (cos 90° + i sin 90°). 
10(cos 180° + i sin 180°). 
V41(cos 309° + i sin 309°). 


6. 7. V2 - iV 2. 9. - .839 + .5450 

13. 2(cos 120° + i sin 120°). 

17. 7(cos 270° + i sin 270°). 

21. 5(cos 307° + i sin 307°). 

25. cos 235° + i sin 235°. 


V2(cos 225° + i sin 225°); conjugate: V2(cos 135° + i sin 135°). 
5V5(cos 117° + i sin 117°); conjugate: 5^5(cos 243° + i sin 243°). 


Exercise 66. Page 167 

- 8 + 8iV§. 3. .00001 (cos 15° + i sin 15°). 5. 3 6 • 2 2 (- 1 + i). 

16 - 16tV3. 9. 5 6 (cos 318° 50' + i sin 318° 50'). 

25 5 (cos 188° 40' + i sin 188° 40'). 

Modulus = 2; amplitudes are 20°, 92°, 164°, 236°, and 308°. 

Modulus = 10; amplitudes are 100°, 220°, and 340°. 

Modulus = 3; amplitudes are 20°, 110°, 200°, and 290°. 

1; M- 1 + iVs); H- 1 ~ tv^). 21. K1 + i'S 3); - 1; |(1 - iV\ 3). 

Modulus = 2; amplitudes are 75°; 165°, 255°, and 345°. 

(1 +JV 3); - 2; (1 - iV 3). 27. .*V5(1 + i); JV$(- 1 - i). 

f(V§ + 0; W ~ V5); - i. 

Modulus = 2; amplitudes are 78f°, 168|°, 258f°, and 348f°. 

(i + VS); 2»; (i - VS); (- i - VS); - 2»; <VS'- i). 

(.766 + .6430; (.174 + .9850; (- .500 + .8660; (- -940 + .3420; 

(- .500 - .8660; (- -940 - .3420; 1; (.174 - .9850; (-766 - .6430- 


Exercise 67. Page 168 

|(eos 45° + i sin 45°). 3. 5(cos 230° -f i sin 230°). 

3(cos 185° + i sin 185°). 7. ^f(cos 120° + % sin 120°). 

cos 50 = cos 6 0 — 10 cos 3 0 sin 2 0 + 5 cos 0 sin 4 0. 
sin 50 = sin 5 0 — 10 cos 2 0 sin 3 0 + 5 cos 4 0 sin 0. 


Exercise 68. Page 169 

The functions are given in the order sine, cosecant, cosine, secant, tangent, and 
cotangent, in Problems 5 and 7. 

6. (- tV, - J+, - tto - ¥)• 7. (0, «, - 1, - 1, 0, =o). 

11. 1.931. 13. .7660. 16. 1.963. 17. - .4226. 


16 


TRIGONOMETRY 


The functions are given in the order sine, tangent, and secant. 

21. (- \Vi, - §V3). 23. (- §V2 ( 1, - Vi). 25. (- Vz, - 2). 


27. 

(a) 

(1^3, 

-VS, - 2); 

(6) (" 

- iV2, 1 

7 

Vi). 




29. 

a = 

.4138; 

; c = .6111; 47° 24'. 

31. 

b = 

66.085; c = 

70.655; 20 c 

’ 43.3', 

33. 

a = 

.4138 

; .41379: c = 

.6114; 

.61132: 

47° 

24'. 




35. 

b = 

661.0 

; 660.86: c = 

706.7; 

706.57: 

20° 

43'; 20° 43. 

3'. 



37. 

30°; 

150°. 

39. 30°; 

210 °. 

41. 

135 

°; 225°. 

43. 

120 °; 

240°. 

45. 

35°; 

325°. 

47. 154° 

40'; 334° 40'. 


49. 45°; 

135°; 

225°; 

315° 

51. 

90°; 

270°. 

53. 90°; 

270°. 



55. 210°; 

330°. 



57. 

60°; 

120 °; 

; 240°; 300°. 


59. 

30°; 

150°; 210° 

; 330° 



65. 

5. 


67. sin 6 . 

69. Sm 

6 — cos 

9 

71. 21 


73. 

2 . 





sin 

8 + cos 

8 





75. 

logiy N = 

5; logio N = 

- .657; 

logs N 

= 1 . 

.32. 




77. 

Mant. = 

.895; char. = 

3: mant. = .9433; 

char. = — 

4: 




mant. = .8505; char. = — 7. 

79. .2367; .23672. . 81. 1.451; 1.4514. 83. - .03928; - .039275. 

85. .006467; .0064680. 87. 3. 95. f$ 7 r; 2.803. 

99. N 25° 56' E. 101. - |(V 2 + ^ 6 ). 103. ^2 + Vs. 

105. - V 3 _ 2 V 2 , or (1 - Vi). 107. V 3 . 

109. The sin, tan, and sec, in this order, are as follows: 

\ a -f p;. v ttY) ttt;> v* P)* v T2T» - stv* 

119. 90°; 270°. 121. 20°; 90°; 100°; 140°; 220°; 260°; 270°; 340°. 123. 135°. 
127. 78° 28'. 129. a = 82° 49'; 0 = 55° 46'; 7 = 41° 25'. 

131. a = 27° 42'; 0 = 83° O'; .3155. 133. 36° 52'. 135. - tan 38°. 

137. - sec 17°. 139. - tan 28°. 141. cos 27°. 143. - cos 15°. 

145. - esc 5°. 147. — cos x. 149. — cot x. 

151. - esc x. 153. 2925. 156. 190.2; 205.9. 

159. (- fV 3 , f); (fV 3 , - -I). 161. (a) 90°, etc.; ( 6 ) 135°, etc.; (c) 60°, etc. 

163. * 7 r. 165. - §tt. 167. - f. 

179. 30°; 60°; 120°; 150°; 210°; 240°; 300°; 330°. 

181. 30°; 210°. 183. No solution. 185. 90°; 180°. 187. - 

189. T V 193. ( 1 ) 2 ; ( 2 ) 25. . 196. 274.2 1b.; N 7° 0.5' E. 

197. 226.6 lb., parallel to plane. 199. 33|. 

201. Xa: .0079706; Xb: .0079706. 203. 19.4 

Exercise 69. Page 184 

1. AD — 1193 yd.; course AD — 1188 yd.; incl. = 5° 8 '. 

3. 329.6 yd.; N 27° 4' W. 





LOGARITHMIC 

AND 

TRIGONOMETRIC TABLES 


COMPILED BY 

WILLIAM L. HART, Ph.D. 

PROFESSOR OF MATHEMATICS 
UNIVERSITY OF MINNESOTA 



D. C. HEATH AND COMPANY 

BOSTON NEW YORK CHICAGO 

ATLANTA SAN FRANCISCO DALLAS 

LONDON 







Copyright, 1933 
By W. L. Hart 


No part of the material covered by this 
copyright may be reproduced in any form 
without written permission of the publisher. 

3 e 3 


PRINTED IN THE UNITED STATES OF AMERICA 



CONTENTS 


TABLE PAGE 

I. SQUARES AND SQUARE ROOTS: 1 — 200 . 1 

II. THREE-PLACE LOGARITHMS OF NUMBERS. 2 

III. THREE-PLACE LOGARITHMS OF THE TRIGONOMETRIC 

FUNCTIONS . 2 

IV. THREE-PLACE VALUES OF THE TRIGONOMETRIC 

FUNCTIONS AND DEGREES IN RADIAN MEASURE 3 

V. FOUR-PLACE LOGARITHMS OF NUMBERS . 4 

VI. FOUR-PLACE LOGARITHMS OF THE TRIGONOMETRIC 

FUNCTIONS .. 6 

VII. FOUR-PLACE VALUES OF THE TRIGONOMETRIC 

FUNCTIONS . 12 

VIII. FIVE-PLACE LOGARITHMS OF NUMBERS . 18 

IX. FIVE-PLACE LOGARITHMS OF THE TRIGONOMETRIC 

FUNCTIONS . 36 

X. AUXILIARY TABLES FOR THE LOGARITHMS OF THE 
TRIGONOMETRIC FUNCTIONS OF ANGLES NEAR TO 0° 

OR 90° . 83 

XI. FIVE-PLACE VALUES OF THE TRIGONOMETRIC FUNC¬ 
TIONS . 88 

XII. SQUARES AND SQUARE ROOTS: 1.00 — 9.99 . 115 

XIII. TABLES INVOLVING RADIAN MEASURE: 

VALUES OF THE TRIGONOMETRIC FUNCTIONS. ... 120 

DEGREES AND MINUTES EXPRESSED IN RADIANS . 121 

XIV. NATURAL OR NAPERIAN LOGARITHMS . 122 

XV. IMPORTANT CONSTANTS AND THEIR LOGARITHMS . 124 

XVI. LOGARITHMS FOR COMPUTING COMPOUND INTEREST 124 


iii 















































•: * . . . 








































I. SQUARES AND SQUARE ROOTS:* 1 — 200 


N 

N 2 

Vn 

N 

N 2 

Vn 

N 

N 2 

Vn 

N 

N 2 

Vn 

1 

1 

1.000 

51 

2,601 

7.141 

101 

10,201 

10.050 

151 

22,801 

12.288 

2 

4 

1.414 

52 

2,704 

7.211 

102 

10,404 

10.100 

152 

23,104 

12.329 

3 

9 

1.732 

53 

2,809 

7.280 

103 

10,609 

10.149 

153 

23,409 

12.369 

4 

16 

2.000 

54 

2,916 

7.348 

104 

10,816 

10.198 

154 

23,716 

12.410 

5 

25 

2.236 

55 

3,025 

7.416 

105 

11,025 

10.247 

155 

24,025 

12.450 

6 

36 

2.449 

56 

3,136 

7.483 

106 

11,236 

10.296 

156 

24,336 

12.490 

7 

49 

2.646 

57 

3,249 

7.550 

107 

11,449 

10.344 

157 

24,649 

12.530 

8 

64 

2.828 

58 

3,364 

7.616 

108 

11,664 

10.392 

158 

24,964 

12.570 

9 

81 

3.000 

59 

3,481 

7.681 

109 

11,881 

10.440 

159 

25,281 

12.610 

10 

100 

3.162 

60 

3,600 

7.746 

110 

12,100 

10.488 

160 

25,600 

12.649 

11 

121 

3.317 

61 

3,721 

7.810 

111 

12,321 

10.536 

161 

25,921 

12.689 

12 

144 

3.464 

62 

3,844 

7.874 

112 

12,544 

10.583 

162 

26,244 

12.728 

13 

169 

3.606 

63 

3,969 

7.937 

113 

12,769 

10.630 

163 

26,569 

12.767 

14 

196 

3.742 

64 

4,096 

8.000 

114 

12,996 

10.677 

164 

26,896 

12.806 

15 

225 

3.873 

65 

4,225 

8.062 

115 

13,225 

10.724 

165 

27,225 

12.845 

16 

256 

4.000 

66 

4,356 

8.124 

116 

13,456 

10.770 

166 

27,556 

12.884 

17 

289 

4.123 

67 

4,489 

8.185 

117 

13,689 

10.817 

167 

27,889 

12.923 

18 

324 

4.243 

68 

4,624 

8.246 

118 

13,924 

10.863 

168 

28,224 

12.962 

19 

361 

4.359 

69 

4,761 

8.307 

119 

14,161 

10.909 

169 

28,561 

13.000 

20 

400 

4.472 

70 

4,900 

8.367 

120 

14,400 

10.954 

170 

28,900 

13.038 

21 

441 

4.583 

71 

5,041 

8.426 

121 

14,641 

11.000 

171 

29,241 

13.077 

22 

484 

4.690 

72 

5,184 

8.485 

122 

14,884 

11.045 

172 

29,584 

13.115 

23 

529 

4.796 

73 

5,329 

8.544 

123 

15,129 

11.091 

173 

29,929 

13.153 

24 

576 

4.899 

74 

5,476 

8.602 

124 

15,376 

11.136 

174 

30,276 

13.191 

25 

625 

5.000 

75 

5,625 

8.660 

125 

15,625 

11.180 

175 

30,625 

13.229 

26 

676 

5.099 

76 

5,776 

8.718 

126 

15,876 

11.225 

176 

30,976 

13.266 

27 

729 

5.196 

77 

5,929 

8.775 

127 

16,129 

11.269 

177 

31,329 

13.304 

28 

784 

5.292 

78 

6,084 

8.832 

128 

16,384 

11.314 

178 

31,684 

13.342 

29 

841 

5.385 

79 

6,241 

8.888 

129 

16,641 

11.358 

179 

32,041 

13.379 

30 

900 

5.477 

80 

6,400 

8.944 

130 

16,900 

11.402 

180 

32,400 

13.416 

31 

961 

5.568 

81 

6,561 

9.000 

131 

17,161 

11.446 

181 

32,761 

13.454 

32 

1,024 

5.657 

82 

6,724 

9.055 

132 

17,424 

11.489 

182 

33,124 

13.491 

33 

1,089 

5.745 

83 

6,889 

9.110 

133 

17,689 

11.533 

183 

33,489 

13.528 

34 

1,156 

5.831 

84 

7,056 

9.165 

134 

17,956 

11.576 

184 

33,856 

13.565 

35 

1,225 

5.916 

85 

7,225 

9.220 

135 

18,225 

11.619 

185 

34,225 

13.601 

36 

1,296 

6.000 

86 

7,396 

9.274 

136 

18,496 

11.662 

186 

34,596 

13.638 

37 

1,369 

6.083 

87 

7,569 

9.327 

137 

18,769 

11.705 

187 

34,969 

13.675 

38 

1,444 

6.164 

88 

7,744 

9.381 

138 

19,044 

11.747 

188 

35,344 

13.711 

39 

1,521 

6.245 

89 

7,921 

9.434 

139 

19,321 

11.790 

189 

35,721 

13.748 

40 

1,600 

6.325 

90 

8,100 

9.487 

140 

19,600 

11.832 

190 

36,100 

13.784 

41 

1,681 

6.403 

91 

8,281 

9.539 

141 

19,881 

11.874 

191 

36,481 

13.820 

42 

1,764 

6.481 

92 

8,464 

9.592 

142 

20,164 

11.916 

192 

36,864 

13.856 

43 

1,849 

6.557 

93 

8,649 

9.644 

143 

20,449 

11.958 

193 

37,249 

13.892 

44 

1,936 

6.633 

94 

8,836 

9.695 

144 

20,736 

12.000 

194 

37,636 

13.928 

45 

2,025 

6.708 

95 

9,025 

9.747 

145 

21,025 

12.042 

195 

38,025 

13.964 

46 

2,116 

6.782 

96 

9,216 

9.798 

146 

21,316 

12.083 

196 

38,416 

14.000 

47 

2,209 

6.856 

97 

9,409 

9.849 

147 

21,609 

12.124 

197 

38,809 

14.036 

48 

2,304 

6.928 

98 

9,604 

9.899 

148 

21,904 

12.166 

198 

39,204 

14.071 

49 

2,401 

7.000 

99 

9,801 

9.950 

149 

22,201 

12.207 

199 

39,601 

14.107 

50 

2,500 

7.071 

100 

10,000 

10.000 

150 

22,500 

12.247 

200 

40,000 

14.142 

N 

N 2 

Vn 

N 

N 2 

Vn 

N 

N 2 

Vn 

N 

N 2 

Vn 


* A more extensive table is found commencing on page 115 


[i] 





































II. THREE-PLACE 
LOGARITHMS 
OF NUMBERS 


III. THREE-PLACE 
LOGARITHMS 
OF FUNCTIONS 



L Sin * 

L Tan * 

L Cot 

L Cos * 


0° 

— 

— 

-- 

10.000 

90° 

1 ° 

8.242 

8.242 

1.758 

10.000 

89 ° 

2 ° 

.543 

.543 

.457 

10.000 

88 ° 

3 ° 

.719 

.719 

.281 

9.999 

87 ° 

4 ° 

.844 

.845 

.155 

.999 

86 ° 

5° 

8.940 

8.942 

1.058 

9.998 

85° 

6 ° 

9.019 

9.022 

0.978 

9.998 

84 ° 

7 ° 

.086 

.089 

.911 

.997 

83 ° 

go 

.144 

.148 

.852 

.996 

82 ° 

9 ° 

.194 

.200 

.800 

.995 

81 ° 

10° 

9.240 

9.246 

0.754 

9.993 

80° 

11 ° 

9.281 

9.289 

0.711 

9.992 

79 ° 

12 ° 

.318 

.327 

.673 

.990 

78 ° 

13 ° 

.352 

.363 

.637 

.989 

77 ° 

14 ° 

.384 

.397 

.603 

.987 

76 ° 

15° 

9.413 

9.428 

0.572 

9.985 

75° 

16 ° 

9.440 

9.458 

0.543 

9.983 

74 ° 

17 ° 

.466 

.485 

.515 

.981 

73 ° 

18 ° 

.490 

.512 

.488 

.978 

72 ° 

19 ° 

.513 

.537 

.463 

.976 

71 ° 

20° 

9.534 

9.561 

0.439 

9.973 

70° 

21 ° 

9.554 

9.584 

0.416 

9.970 

69 ° 

22 ° 

.574 

.606 

.394 

.967 

68 ° 

23 ° 

.592 

.628 

.372 

.964 

67 ° 

24 ° 

.609 

.649 

.351 

.961 

66 ° 

25° 

9.626 

9.669 

0.331 

9.957 

65° 

26 ° 

9.642 

9.688 

0.312 

9.954 

64 ° 

27 ° 

.657 

.707 

.293 

.950 

63 ° 

28 ° 

.672 

.726 

.274 

.946 

62 ° 

29 ° 

.686 

.744 

.256 

.942 

61 ° 

30° 

9.699 

9.761 

0.239 

9.938 

60° 

31 ° 

9.712 

9.779 

0.221 

9.933 

59 ° 

32 ° 

.724 

.796 

.204 

.928 

58 ° 

33 ° 

.736 

.813 

.187 

.924 

57 ° 

34 ° 

.748 

.829 

.171 

.919 

56 ° 

35° 

9.759 

9.845 

0.155 

9.913 

55° 

36 ° 

9.769 

9.861 

0.139 

9.908 

54 ° 

37 ° 

.779 

.877 

.123 

.902 

53 ° 

38 ° 

.789 

.893 

.107 

.897 

52 ° 

39 ° 

.799 

.908 

.092 

.891 

51 ° 

40° 

9.808 

9.924 

0.076 

9.884 

50° 

41 ° 

9.817 

9.939 

0.061 

9.878 

49 ° 

42 ° 

.826 

.954 

.046 

.871 

48 ° 

43 ° 

.834 

.970 

.030 

.864 

47 ° 

44 ° 

.842 

.985 

.015 

.857 

46 ° 

45° 

9.849 

10.000 

0.000 

9.849 

45° 


L Cos * 

L Cot * 

L Tan 

L Sin * 



N 

Log N 

N 

Log N 

1.0 

.000 

5.5 

.740 

1.1 

.041 

5.6 

.748 

1.2 

.079 

5.7 

.756 

1.3 

.114 

5.8 

.763 

1.4 

.146 

5.9 

.771 

1.5 

.176 

6.0 

.778 

1.6 

.204 

6.1 

.785 

1.7 

.230 

6.2 

.792 

1.8 

.255 

6.3 

.799 

1.9 

.279 

6.4 

.806 

2.0 

.301 

6.5 

.813 

2.1 

.322 

6.6 

.820 

2.2 

.342 

6.7 

.826 

2.3 

.362 

6.8 

.833 

2.4 

.380 

6.9 

.839 

2.5 

.398 

7.0 

.845 

2.6 

.415 

7.1 

.851 

2.7 

.431 

7.2 

.857 

2.8 

.447 

7.3 

.863 

2.9 

.462 

7.4 

.869 

3.0 

.477 

7.5 

.875 

3.1 

.491 

7.6 

.881 

3.2 

.505 

7.7 

.886 

3.3 

.519 

7 . 8 . 

.892 

3.4 

.531 

7.9 

.898 

3.5 

.544 

8.0 

.903 

3.6 

.556 

8.1 

.908 

3.7 

.568 

8.2 

.914 

3.8 

.580 

8.3 

.919 

3.9 

.591 

8.4 

.924 

4.0 

.602 

8.5 

.929 

4.1 

.613 

8.6 

.935 

4.2 

.623 

8.7 

.940 

4.3 

.633 

8.8 

.944 

4.4 

.643 

8.9 

.949 

4.5 

.653 

9.0 

.954 

4.6 

.663 

9.1 

.959 

4.7 

.672 

9.2 

.964 

4.8 

.681 

9.3 

.968 

4.9 

.690 

9.4 

.973 

5.0 

.699 

9.5 

.978 

5.1 

.708 

9.6 

.982 

5.2 

.716 

9.7 

.987 

5.3 

.724 

9.8 

.991 

5.4 

.732 

9.9 

.996 

5.5 

.740 

1.00 

1.000 

N 

Log N 

N 

Log N 


[2] 


* Subtract 10 from each entry in this column. 
















































IV. THREE-PLACE VALUES OF TRIGONOMETRIC FUNCTIONS 

AND 

DEGREES IN RADIAN MEASURE 


Rad. 

Deg. 

Sin 

Tan 

Sec 

Csc 

Cot 

Cos 

Deg. 

Rad. 

.000 

0° 

.000 

.000 

1.000 

— 

— 

1.000 

90° 

1.571 

.017 

1° 

.017 

.017 

1.000 

57.30 

57.29 

1.000 

89° 

1.553 

.035 

2° 

.035 

.035 

1.001 

28.65 

28.64 

0.999 

88° 

1.536 

.052 

3° 

.052 

.052 

1.001 

19.11 

19.08 

.999 

87° 

1.518 

.070 

4 ° 

.070 

.070 

1.002 

14.34 

14.30 

.998 

86° 

1.501 

.087 

5° 

.087 

.087 

1.004 

11.47 

11.43 

.996 

85° 

1.484 

.105 

6° 

.105 

.105 

1.006 

9.567 

9.514 

.995 

84° 

1.466 

.122 

7° 

.122 

.123 

1.008 

8.206 

8.144 

.993 

83° 

1.449 

.140 

8° 

.139 

.141 

1.010 

7.185 

7.115 

.990 

82° 

1.431 

.157 

9° 

.156 

.158 

1.012 

6.392 

6.314 

.988 

81° 

1.414 

.175 

10° 

.174 

.176 

1.015 

5.759 

5.671 

.985 

80° 

1.396 

.192 

11° 

.191 

.194 

1.019 

5.241 

5.145 

.982 

79° 

1.379 

.209 

12° 

.208 

.213 

1.022 

4.810 

4.705 

.978 

78° 

1.361 

.227 

13° 

.225 

.231 

1.026 

4.445 

4.331 

.974 

77° 

1.344 

.244 

14° 

.242 

.249 

1.031 

4.134 

4.011 

.970 

76° 

1.326 

.262 

15° 

.259 

.268 

1.035 

3.864 

3.732 

.966 

75° 

1.309 

.279 

16° 

.276 

.287 

1.040 

3.628 

3.487 

.961 

74° 

1.292 

.297 

17° 

.292 

.306 

1.046 

3.420 

3.271 

, 9£6 

73° 

1.274 

.314 

18° 

.309 

.325 

1.051 

3.236 

3 . 078 . 

.951 

72° 

1.257 

.332 

19° 

.326 

.344 

1.058 

3.072 

2.904 

.946 

71° 

1.239 

.349 

20° 

.342 

.364 

1.064 

2.924 

2.747 

.940 

70° 

1.222 

.367 

21° 

.358 

.384 

1.071 

2.790 

2.605 

.934 

69° 

1.204 

.384 

22° 

.375 

.404 

1.079 

2.669 

2.475 

.927 

68° 

1.187 

.401 

23° 

.391 

.424 

1.086 

2.559 

2.356 

.921 

67° 

1.169 

.419 

24° 

.407 

.445 

1.095 

2.459 

2.246 

.914 

66° 

1.152 

.436 

25° 

.423 

.466 

1.103 

2.366 

2.145 

.906 

65° 

1.134 

.454 

26° 

.438 

.488 

1.113 

2.281 

2.050 

.899 

64° 

1.117 

.471 

27° 

.454 

.510 

1.122 

2.203 

1.963 

.891 

63° 

1.100 

.489 

28° 

.469 

.532 

1.133 

2.130 

1.881 

.883 

62° 

1.082 

.506 

29° 

.485 

.554 

1.143 

2.063 

1.804 

.875 

61° 

1.065 

.524 

30° 

.500 

.577 

1.155 

2.000 

1.732 

.866 

60° 

1.047 

.541 

31° 

.515 

.601 

1.167 

1.942 

1.664 

.857 

59° 

1.030 

.559 

32° 

.530 

.625 

1.179 

1.887 

1.600 

.848 

58° 

1.012 

.576 

33° 

.545 

.649 

1.192 

1.836 

1.540 

.839 

57° 

0.995 

.593 

34° 

.559 

.675 

1.206 

1.788 

1.483 

.829 

56° 

0.977 

.611 

35° 

.574 

.700 

1.221 

1.743 

1.428 

.819 

55° 

0.960 

.628 

36° 

.588 

.727 

1.236 

1.701 

1.376 

.809 

54° 

0.942 

.646 

37° 

.602 

.754 

1.252 

1.662 

1.327 

.799 

53° 

0.925 

.663 

38° 

.616 

.781 

1.269 

1.624 

1.280 

.788 

52° 

0.908 

.681 

39° 

.629 

.810 

1.287 

1.589 

1.235 

.777 

51° 

0.890 

.698 

40° 

.643 

.839 

1.305 

1.556 

1.192 

.766 

50° 

0.873 

.716 

41° 

.656 

.869 

1.325 

1.524 

1.150 

.755 

49° 

0.855 

.733 

42° 

.669 

.900 

1.346 

1.494 

1.111 

.743 

48° 

0.838 

.750 

43° 

.682 

.933 

1.367 

1.466 

1.072 

.731 

47° 

0.820 

.768 

44° 

.695 

0.966 

1.390 

1.440 

1.036 

.719 

46° 

0.803 

.785 

45° 

.707 

1.000 

1.414 

1.414 

1.000 

.707 

45° 

0.785 

Rad. 

Deg. 

Cos 

Cot 

Csc 

Sec 

Tan 

Sin 

Deg. 

Rad. 


[3] 




















































V. FOUR-PLACE LOGARITHMS OF NUMBERS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 

10 

.0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 





11 

.0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 





12 

.0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 





13 

.1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 


28 

27 

26 

14 

.1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

1 

2 8 

2 7 

2 6 

15 

.1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

2 

5.6 

5.4 

5.2 

16 

.2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

3 

8.4 

8.1 

7.8 

17 

.2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

4 

5 

11.2 
14 0 

10.8 
13 5 

10.4 
13 0 

18 

.2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

6 

16.8 

16.2 

15.6 

19 

.2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

7 

19.6 

18.9 

18.2 












8 

22.4 

21.6 

20.8 

20 

.3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

9 

25.2 

24.3 

23.4 

21 

.3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 





22 

.3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 





23 

.3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 


22 

21 

20 

24 

.3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

1 

2.2 

2.1 

2.0 

25 

.3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

z 

3 

4.4 

6 6 

4.2 

6 3 

4.U 
6 0 

26 

.4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

4 

8.8 

8.4 

8.0 

27 

.4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

5 

11.0 

10.5 

10.0 

28 

.4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

6 

7 

13.2 

12.6 
14 7 

12.0 
14 0 

29 

.4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

8 

17.6 

16.8 

16.0 












9 

19.8 

18.9 

18.0 

30 

.4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 





31 

.4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 





32 

.5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 


16 

15 

14 

33 

.5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

1 

1.6 

1.5 

1.4 

34 

.5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

2 

3.2 

3.0 

2.8 

35 

.5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539' 

5551 

3 

4.8 

4.5 

4.2 

36 

.5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

4 

5 

6.4 

8.0 

6.0 

7.5 

5.6 

7.0 

37 

.5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

6 

9.6 

9.0 

8.4 

38 

.5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

7 

11.2 

10.5 

9.8 

39 

.5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

8 

12.8 

12.0 

11.2 












9 

14.4 

13.5 

12.6 

40 

.6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 





41 

.6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 


13 

12 

11 

42 

.6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

— 




43 

.6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

1 

1.3 

1.2 

1.1 












2 

2.6 

2.4 

2.2 

44 

.6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

3 

3.9 

3.6 

3.3 

45 

.6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

4 

5.2 

4.8 

4.4 

46 

.6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

5 

6.5 

6.0 

5.5 












6 

7.8 

7.2 

6.6 

47 

.6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

7 

9.1 

8.4 

7.7 

48 

.6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

8 

10.4 

9.6 

8.8 

49 

.6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

9 

11.7 

10.8 

9.9 

50 

.6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 





N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 







43 

42 

41 

40 

39 


38 

37 

36 

35 

34 


33 

32 

31 

30 

29 


1 

4.3 

4.2 

4.1 

4.0 

3.9 

1 

3.8 

3.7 

3.6 

3.5 

3.4 

1 

3.3 

3.2 

3.1 

3.0 

2.9 

1 

2 

8.6 

8.4 

8.2 

8.0 

7.8 

2 

7.6 

7.4 

7.2 

7.0 

6.8 

2 

6.6 

6.4 

6.2 

6.0 

5.8 

2 

3 

12.9 

12.6 

12.3 

12.0 

11.7 

3 

11.4 

11.1 

10.8 

10.5 

10.2 

3 

9.9 

9.6 

9.3 

9.0 

8.7 

3 

4 

17.2 

16.8 

16.4 

16.0 

15.6 

4 

15.2 

14.8 

14.4 

14.0 

13.6 

4 

13.2 

12.8 

12.4 

12.0 

11.6 

4 

5 

21.5 

21.0 

20.5 

20.0 

19.5 

5 

19.0 

18.5 

18.0 

17.5 

17.0 

5 

16.5 

16.0 

15.5 

15.0 

14.5 

5 

6 

25.8 

25.2 

24.6 

24.0 

23.4 

6 

22.8 

22.2 

21.6 

21.0 

20.4 

6 

19.8 

19.2 

18.6 

18.0 

17.4 

6 

7 

30.1 

29.4 

28.7 

28.0 

27.3 

7 

26.6 

25.9 

25.2 

24.5 

23.8 

7 

23.1 

22.4 

21.7 

21.0 

20.3 

7 

8 

34.4 

33.6 

32.8 

32.0 

31.2 

8 

30.4 

29.6 

28.8 

28.0 

27.2 

8 

26.4 

25.6 

24.8 

24.0 

23.2 

8 

9 

38.7 

37.8 

36.9 

36.0 

35.1 

9 

34.2 

33.3 

32.4 

31.5 

30.6 

9 

29.7 

28.8 

27.9 

27.0 

26.1 

9 


[4] 








































































































































































V. FOUR-PLACE LOGARITHMS OF NUMBERS 


Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 







50 

.6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 







51 

.7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 







52 

.7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 


25 

24 

23 

53 

.7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

1 

2.5 

2.4 

2.3 

54 

.7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

2 

5.0 

4.8 

4.6 

55 

.7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

3 

4 

7.5 

10.0 

7.2 

9.6 

6.9 

9.2 

56 

.7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

5 

12.5 

12.0 

11.5 

57 

.7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

6 

15.0 

14.4 

13.8 

58 

.7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

7 

8 

17.5 

20.0 

16.8 

19.2 

16.1 

18.4 

59 

.7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

9 

22.5 

21.6 

20.7 

60 

.7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 







61 

.7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 


19 

18 

17 

62 

.7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

— 






63 

.7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

1 

1.9 

1.8 

1.7 












2 

3.8 

3.6 

3.4 

64 

.8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

3 

5.7 

5.4 

5.1 

65 

.8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

4 

7.6 

7.2 

6.8 

66 

.8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

5 

9.5 

9.0 

8.5 












6 

11.4 

10.8 

10.2 

67 

.8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

7 

13.3 

12.6 

11.9 

68 

.8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

8 

15.2 

14.4 

13.6 

69 

.8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

9 

17 1 

10 2 

15 3 


















70 

.8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 



in 


9 

71 

.8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 







72 

.8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 


1 


1.0 


0.9 

73 

.8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 


2 

3 

z.u 

3.0 


1.8 

2.7 

74 

.8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 


4 

4.0 


3.6 

75 

.8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 


5 


5.0 


4.5 

76 

.8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 


6 

7 

b.U 
7 0 


5.4 

6.3 

77 

.8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 


8 


8.0 


7.2 

78 

.8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 


9 


9.0 


8.1 

79 

.8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 







80 

.9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 




8 


7 

81 

.9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 


1 

0 8 

0 7 

82 

.9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 


2 

1.6 

1.4 

83 

.9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 


3 

A 

2.4 

o 9 

2.1 

84 

.9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 


4 

5 

6.Z 

4 0 

4.0 

3 5 

85 

.9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 


6 

4.8 

4.2 

86 

.9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 


7 

& 

5.6 

6 4 

4.9 

87 

.9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 


9 

7 2 

6 3 

88 

.9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 







89 

.9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 







90 

.9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 


6 


5 


4 












— 






91 

.9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

1 

o 

U.b 

1 9 

U.b 
i n 

0.4 

n q 

92 

.9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

A 

3 

JL.Z 

1.8 

i.U 

1.5 

U.o 

1.2 

93 

.9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

4 

2.4 

2.0 

1.6 

94 

.9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

0 

6 

o.U 

3 6 

4.0 

3 0 

4.U 

2 4 

95 

.9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

7 

4.2 

3.5 

2.8 

96 

.9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

8 

q 

4.8 
e; a 

4.0 

A K 

3.2 

2 6 

97 

.9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

«7 


'i.O 


98 

.9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 







99 

.9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 







N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


C5] 















































































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 0° — 6°; 84° —90° 


0 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
0 ° 50 ' 

1 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
1 ° 50 ' 

2 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
2 ° 50 ' 

3° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
3 ° 50 ' 

4° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
4 ° 50 ' 
5° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
5 ° 50 ' 
6 ° 00 ' 


7.4637 

.7648 

7.9408 

8.0658 

.1627 


8.2419 


.3088 

.3668 

.4179 

.4637 

.5050 


8.5428 


.5776 

.6097 

.6397 

.6677 

.6940 


8.7188 


.7423 

.7645 

.7857 

.8059 

.8251 


8.8436 


.8613 

.8783 

.8946 

.9104 

.9256 


8.9403 


.9545 

.9682 

.9816 

8.9945 

9.0070 


9.0192 


L Cos* 


222 

212 

202 

192 

185 

177 

170 

163 

158 

152 

147 

142 

137 

134 

129 

125 

122 


7.4637 

.7648 

7.9409 

8.0658 

.1627 


8.2419 


.3089 

.3669 

.4181 

.4638 

.5053 


8.5431 


.5779 

.6101 

.6401 

.6682 

.6945 


8.7194 


.7429 

.7652 

.7865 

.8067 

.8261 


8.8446 


.8624 

.8795 

.8960 

.9118 

.9272 


8.9420 


.9563 

.9701 

.9836 

8.9966 

9.0093 


9.0216 


223 

213 

202 

194 

185 

178 

171 

165 

158 

154 

148 

143 

138 

135 

130 

127 

123 


L Cot 

L Cos* 



10.0000 

90° 00' 

2.5363 

.0000 

89 ° 50 ' 

.2352 

.0000 

40 ' 

2.0591 

.0000 

30 ' 

1.9342 

.0000 

20 ' 

.8373 

10.0000 

10 ' 

1.7581 

9.9999 

89° 00' 

.6911 

.9999 

88 ° 50 ' 

.6331 

.9999 

40 ' 

.5819 

.9999 

30 ' 

.5362 

.9998 

20 ' 

.4947 

.9998 

10 ' 

1.4569 

9.9997 

88° 00' 

.4221 

.9997 

87 ° 50 ' 

.3899 

.9996 

40 ' 

.3599 

.9996 

30 ' 

.3318 

.9995 

20 ' 

.3055 

.9995 

10 ' 

1.2806 

9.9994 

87° 00' 

.2571 

.9993 

86 ° 50 ' 

.2348 

.9993 

40 ' 

.2135 

.9992 

30 ' 

.1933 

.9991 

20 ' 

.1739 

.9990 

10 ' 

1.1554 

9.9989 

86° 00' 

.1376 

.9989 

85 ° 50 ' 

.1205 

.9988 

40 ' 

.1040 

.9987 

30 ' 

.0882 

.9986 

20 ' 

.0728 

.9985 

10 ' 

1.0580 

9.9983 

85° 00' 

.0437 

.9982 

84 ° 50 ' 

.0299 

.9981 

40 ' 

.0164 

.9980 

30 ' 

1.0034 

.9979 

20 ' 

0.9907 

.9977 

10 ' 

0.9784 

9.9976 

84° 00' 

L Tan 

L Sin* 



Prop. Parts 


To avoid interpolating, 
for angles between 0° and 
3 °, or between 87 ° and 90 °, 
use Table IX. 



2 

223 

222 

213 

1 

0.2 

22 

22 

21 

2 

0.4 

45 

44 

43 

3 

0.6 

67 

67 

64 

4 

0.8 

89 

89 

85 

6 

1.0 

112 

111 

106 

6 

1.2 

134 

133 

128 

7 

1.4 

156 

155 

149 

8 

1.6 

178 

178 

170 

9 

1.8 

201 

200 

192 



3 

212 

202 

194 

1 

0.3 

21 

20 

19 

2 

0.6 

42 

40 

39 

3 

0.9 

64 

61 

58 

4 

1.2 

85 

81 

78 

5 

1.5 

106 

101 

97 

6 

1.8 

127 

121 

116 

7 

2.1 

148 

141 

136 

8 

2.4 

170 

162 

155 

9 

2.7 

191 

182 

175 



192 

185 

178 

1 

19 

18 

18 

2 

38 

37 

36 

3 

58 

56 

53 

4 

77 

74 

71 

5 

96 

92 

89 

6 

115 

111 

107 

7 

134 

130 

125 

8 

154 

148 

142 

9 

173 

166 

160 



177 

171 

170 

165 

163 


158 

1 

18 

17 

17 

16 

16 

1 

16 

2 

35 

34 

34 

33 

33 

2 

32 

3 

53 

51 

51 

50 

49 

3 

47 

4 

71 

68 

68 

66 

65 

4 

63 

5 

88 

86 

85 

82 

82 

5 

79 

6 

106 

103 

102 

99 

98 

6 

95 

7 

124 

120 

119 

116 

114 

7 

111 

8 

142 

137 

136 

132 

130 

8 

126 

9 

159 

154 

153 

148 

147 

9 

142 


134 

130 

129 

127 

125 


123 

1 

13 

13 

13 

13 

12 

1 

12 

2 

27 

26 

26 

25 

25 

2 

25 

3 

40 

39 

39 

38 

38 

3 

37 

4 

54 

52 

52 

51 

50 

4 

49 

5 

67 

65 

64 

64 

62 

5 

62 

6 

80 

78 

77 

76 

75 

6 

74 

7 

94 

91 

90 

89 

88 

7 

86 

8 

107 

104 

103 

102 

100 

8 

98 

9 

121 

117 

116 

114 

112 

9 

111 


154 


15 

31 

46 

62 

77 

92 

108 

123 

139 

122 


152 


12 

24 

37 

49 

61 

73 

85 

98 

110 


15 

30 

46 

61 

76 

91 

106 

122 

137 

120 


148 


12 

24 

36 

48 

60 

72 

84 

96 

108 


15 

30 

44 

59 

74 

89 

104 

118 

133 

119 


12 

24 

36 

48 

60 

71 

83 

95 

107 


147 


143 

142 

138 

137 

135 


15 

1 

14 

14 

14 

14 

14 

1 

29 

2 

29 

28 

28 

27 

27 

2 

44 

3 

43 

43 

41 

41 

40 

3 

59 

4 

57 

57 

55 

55 

54 

4 

74 

5 

72 

71 

69 

68 

68 

5 

88 

6 

86 

85 

83 

82 

81 

6 

103 

7 

100 

99 

97 

96 

94 

7 

118 

8 

114 

114 

110 

110 

108 

8 

132 

9 

129 

128 

124 

123 

122 

9 

117 


115 

114 

113 

111 

109 


12 

1 

12 

11 

11 

11 

11 

1 

23 

2 

23 

23 

23 

22 

22 

2 

35 

3 

34 

34 

34 

33 

33 

3 

47 

4 

46 

46 

45 

44 

44 

4 

58 

5 

58 

57 

56 

56 

54 

5 

70 

6 

69 

68 

68 

67 

65 

6 

82 

7 

80 

80 

79 

78 

76 

7 

94 

8 

92 

91 

90 

89 

87 

8 

105 

9 

104 

103 

102 

100 

98 

9 


[6] 











































































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 6° — 12°; 78° — 84° 


Prop. Parts 


Subtract 10 from each 
entry in the columns 
marked with through¬ 
out the table. 



108 

107 

105 

1 

10.8 

10.7 

10.5 

2 

21.6 

21.4 

21.0 

3 

32.4 

32.1 

31.5 

4 

43.2 

42.8 

42.0 

6 

54.0 

53.5 

52.5 

6 

64.8 

64.2 

63.0 

7 

75.6 

74.9 

73.5 

8 

86.4 

85.6 

84.0 

9 

97.2 

96.3 

94.5 



104 

102 

101 

1 

10.4 

10.2 

10.1 

2 

20.8 

20.4 

20.2 

3 

31.2 

30.6 

30.3 

4 

41.6 

40.8 

40.4 

5 

52.0 

51.0 

50.5 

6 

62.4 

61.2 

60.6 

7 

72.8 

71.4 

70.7 

8 

83.2 

81.6 

80.8 

9 

93.6 

91.8 

90.9 



99 

98 

97 

95 

1 

9.9 

9.8 

9.7 

9.5 

2 

19.8 

19.6 

19.4 

19.0 

3 

29.7 

29.4 

29.1 

28.5 

4 

39.6 

39.2 

38.8 

38.0 

5 

49.5 

49.0 

48.5 

47.5 

6 

59.4 

58.8 

58.2 

57.0 

7 

69.3 

68.6 

67.9 

66.5 

8 

79.2 

78.4 

77.6 

76.0 

9 

89.1 

88.2 

87.3 

85.5 


6 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
6 ° 50 ' 

7° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
7 ° 50 ' 

8 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
8 ° 50 ' 

9° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
9 ° 50 ' 

10 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
10 ° 50 ' 

11 ° 00 ' 

10 ' 
20 ' 
30 ' 
40 ' 
11 ° 50 ' 

12 ° 00 ' 


L Sin* 


9.0192 


.0311 

.0426 

.0539 

.0648 

.0755 


9.0859 


.0961 

.1060 

.1157 

.1252 

.1345 


9.1436 


.1525 

.1612 

.1697 

.1781 

.1863 


9.1943 


.2022 

.2100 

.2176 

.2251 

.2324 


9.2397 


.2468 

.2538 

.2606 

.2674 

.2740 


9.2806 


.2870 

.2934 

.2997 

.3058 

.3119 


9.3179 


L Cos* 


119 

115 

113 

109 

107 

104 

102 

99 

97 

95 

93 

91 

89 

87 

85 

84 

82 

80 

79 

78 

76 

75 

73 

73 

71 

70 

68 

68 

66 

66 

64 

64 

63 

61 

61 

60 


L Tan* c d 


9.0216 


.0336 

.0453 

.0567 

.0678 

.0786 


9.0891 


.0995 

.1096 

.1194 

.1291 

.1385 


9.1478 


.1569 

.1658 

.1745 

.1831 

.1915 


9.1997 


.2078 

.2158 

.2236 

.2313 

.2389 


9.2463 


.2536 

.2609 

.2680 

.2750 

.2819 


9.2887 


.2953 

.3020 

.3085 

.3149 

.3212 


9.3275 


120 

117 

114 

111 

108 

105 

104 

101 

98 

97 

94 

93 

91 

89 

87 

86 

84 

82 

81 

80 

78 

77 

76 

74 

73 

73 

71 

70 

69 

68 


L Cot 

L Cos* 


0.9784 

9.9976 

84° 00' 

.9664 

.9975 

83 ° 50 ' 

.9547 

.9973 

40 ' 

.9433 

.9972 

30 ' 

.9322 

.9971 

20 ' 

.9214 

.9969 

10 ' 

0.9109 

9.9968 

83° 00' 

.9005 

.9966 

82 ° 50 ' 

.8904 

.9964 

40 ' 

.8806 

.9963 

30 ' 

.8709 

.9961 

20 ' 

.8615 

.9959 

10 ' 

0.8522 

9.9958 

82° 00' 

.8431 

.9956 

81 ° 50 ' 

.8342 

.9954 

40 ' 

.8255 

.9952 

30 ' 

.8169 

.9950 

20 ' 

.8085 

.9948 

10 ' 

0.8003 

9.9946 

81° 00' 

.7922 

.9944 

80 ° 50 ' 

.7842 

.9942 

40 ' 

.7764 

.9940 

30 ' 

.7687 

.9938 

20 ' 

.7611 

.9936 

10 ' 

0.7537 

9.9934 

80° 00' 

.7464 

.9931 

79 ° 50 ' 

.7391 

.9929 

40 ' 

.7320 

.9927 

30 ' 

.7250 

.9924 

20 ' 

.7181 

.9922 

10 ' 

0.7113 

9.9919 

79° 00' 

.7047 

.9917 

78 ° 50 ' 

.6980 

.9914 

40 ' 

.6915 

.9912 

30 ' 

.6851 

.9909 

20 ' 

.6788 

.9907 

10 ' 

0.6725 

9.9904 

78° 00' 

L Tan 

L Sin* 





94 

93 

91 

89 


87 

86 

85 

84 

82 


81 

80 

79 

78 

77 


1 

9.4 

9.3 

9.1 

8.9 

1 

8.7 

8.6 

8.5 

8.4 

8.2 

1 

8.1 

8 

7.9 

7.8 

7.7 

1 

2 

18.8 

18.6 

18.2 

17.8 

2 

17.4 

17.2 

17.0 

16.8 

16.4 

2 

16.2 

16 

15.8 

15.6 

15.4 

2 

3 

28.2 

27.9 

27.3 

26.7 

3 

26.1 

25.8 

25.5 

25.2 

24.6 

3 

24.3 

24 

23.7 

23.4 

23.1 

3 

4 

37.6 

37.2 

36.4 

35.6 

4 

34.8 

34.4 

34.0 

33.6 

32.8 

4 

32.4 

32 

31.6 

31.2 

30.8 

4 

5 

47.0 

46.5 

45.5 

44.5 

5 

43.5 

43.0 

42.5 

42.0 

41.0 

5 

40.5 

40 

39.5 

39.0 

38.5 

5 

6 

56.4 

55.8 

54.6 

53.4 

6 

52.2 

51.6 

51.0 

50.4 

49.2 

6 

48.6 

48 

47.4 

46.8 

46.2 

6 

7 

65.8 

65.1 

63.7 

62.3 

7 

60.9 

60.2 

59.5 

58.8 

57.4 

7 

56.7 

56 

55.3 

54.6 

53.9 

7 

8 

75.2 

74.4 

72.8 

71.2 

8 

69.6 

68.8 

68.0 

67.2 

65.6 

8 

64.8 

64 

63.2 

62.4 

61.6 

8 

9 

84.6 

83.7 

81.9 

80.1 

9 

78.3 

77.4 

76.5 

75.6 

73.8 

9 

72.9 

72 

71.1 

70.2 

69.3 

9 



76 

75 

74 

73 

71 


70 

69 

68 

67 

66 


65 

64 

63 

61 

60 


1 

7.6 

7.5 

7.4 

7.3 

7.1 

1 

7 

6.9 

6.8 

6.7 

6.6 

1 

6.5 

6.4 

6.3 

6.1 

6 

1 

2 

15.2 

15.0 

14.8 

14.6 

14.2 

2 

14 

13.8 

13.6 

13.4 

13.2 

2 

13.0 

12.8 

12.6 

12.2 

12 

2 

3 

22.8 

22.5 

22.2 

21.9 

21.3 

3 

21 

20.7 

20.4 

20.1 

19.8 

3 

19.5 

19.2 

18.9 

18.3 

18 

3 

4 

30.4 

30.0 

29.6 

29.2 

28.4 

4 

28 

27.6 

27.2 

26.8 

26.4 

4 

26.0 

25.6 

25.2 

24.4 

24 

4 

5 

38.0 

37.5 

37.0 

36.5 

35.5 

5 

35 

34.5 

34.0 

33.5 

33.0 

5 

32.5 

32.0 

31.5 

30.5 

30 

5 

6 

45.6 

45.0 

44.4 

43.8 

42.6 

6 

42 

41.4 

40.8 

40.2 

39.6 

6 

39.0 

38.4 

37.8 

36.6 

36 

6 

7 

53.2 

52.5 

51.8 

51.1 

49.7 

7 

49 

48.3 

47.6 

46.9 

46.2 

7 

45.5 

44.8 

44.1 

42.7 

42 

7 

8 

60.8 

60.0 

59.2 

58.4 

56.8 

8 

56 

55.2 

54.4 

53.6 

52.8 

8 

52.0 

51.2 

50.4 

48.8 

48 

8 

9 

68.4 

67.5 

66.6 

65.7 

63.9 

9 

63 

62.1 

61.2 

60.3 

59.4 

9 

58.5 

57.6 

56.7 

54.9 

54 

9 


[7] 










































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 12° —19 6 ; 71°—78" 


12 ° 00 ' 

10 ' 
20 ' 
30' 
40' 
12° 50' 

13° 00' 

10 ' 
20 ' 
30' 
40' 
13° 50' 

14° 00' 

10 ' 
20 ' 
30' 
40' 
14° 50' 

15° 00' 

10 ' 
20 ' 
30' 
40' 
15° 50' 

16° 00' 

10 ' 
20 ' 
30' 
40' 
16° 50' 

17° 00' 

10 ' 
20 ' 
30' 
40' 
17° 50' 
18° 00' 

10 ' 
20 ' 
30' 
40' 
18° 50' 

19° 00' 


L Sin* 


9.3179 


.3238 

.3296 

.3353 

.3410 

.3466 


9.3521 


.3575 

.3629 

.3682 

.3734 

.3786 


9.3837 


.3887 

.3937 

.3986 

.4035 

.4083 


9.4130 


.4177 

.4223 

.4269 

.4314 

.4359 


9.4403 


.4447 

.4491 

.4533 

.4576 

.4618 


9.4659 


.4700 

.4741 

.4781 

.4821 

.4861 


9.4900 


.4939 

.4977 

.5015 

.5052 

.5090 


9.5126 


L Cos * 


59 

58 

57 

57 

56 

55 

54 

54 

53 

52 

52 

51 

50 

50 

49 

49 

48 

47 

47 

46 

46 

45 

45 

44 

44 

44 

42 

43 
42 

41 

41 

41 

40 

40 

40 

39 

39 

38 

38 

37 

38 

36 


9.3275 


.3336 

.3397 

.3458 

.3517 

.3576 


9.3634 


.3691 

.3748 

.3804 

.3859 

.3914 


9.3968 


.4021 

.4074 

.4127 

.4178 

.4230 


9.4281 


.4331 

.4381 

.4430 

.4479 

.4527 


9.4575 


.4622 

.4669 

.4716 

.4762 

.4808 


9.4853 


.4898 

.4943 

.4987 

.5031 

.5075 


9.5118 


.5161 

.5203 

.5245 

.5287 

.5329 


9.5370 


61 

61 

61 

59 

59 

58 

57 

57 

56 

55 

55 

54 

53 

53 

53 

51 

52 

51 

50 

50 

49 

49 

48 

48 

47 

47 

47 

46 

46 

45 

45 

45 

44 

44 

44 

43 

43 

42 

42 

42 

42 

41 


L Cot 

L Cos* 


0.6725 

9.9904 

78° 00' 

.6664 

.9901 

77° 50' 

.6603 

.9899 

40' 

.6542 

.9896 

30' 

.6483 

.9893 

20' 

.6424 

.9890 

10' 

0.6366 

9.9887 

<1 

O 

O 

© 

.6309 

.9884 

76° 50' 

.6252 

.9881 

40' 

.6196 

.9878 

30' 

.6141 

.9875 

20' 

.6086 

.9872 

10' 

0.6032 

9.9869 

o 

o 

o 

© 

.5979 

.9866 

75° 50' 

.5926 

.9863 

40' 

.5873 

.9859 

30' 

.5822 

.9856 

20' 

.5770 

.9853 

10' 

0.5719 

9.9849 

75° 00' 

* .5669 

.9846 

74° 50' 

.5619 

.9843 

40' 

.5570 

.9839 

30' 

.5521 

.9836 

20' 

.5473 

.9832 

10' 

0.5425 

9.9828 

b 

o 

o 

.5378 

.9825 

73° 50' 

.5331 

.9821 

40' 

.5284 

.9817 

30' 

.5238 

.9814 

20' 

.5192 

.9810 

10' 

0.5147 

9.9806 

73° 00' 

.5102 

.9802 

72° 50' 

.5057 

.9798 

40' 

.5013 

.9794 

30' 

.4969 

.9790 

20' 

.4925 

.9786 

10' 

0.4882 

9.9782 

72° 00' 

.4839 

.9778 

71° 50' 

.4797 

.9774 

40' 

.4755 

.9770 

30' 

.4713 

.9765 

20' 

.4671 

.9761 

10' 

0.4630 

9.9757 

•3 

O 

o 

© 

L Tan 

L Sin* 

-< - 


Prop. Parts 


5.1 

10.2 

15.3 

20.4 

25.5 

30.6 

35.7 

40.8 

45.9 



61 

59 

58 

1 

6.1 

5.9 

5.8 

2 

12.2 

11.8 

11.6 

3 

18.3 

17.7 

17.4 

4 

24.4 

23.6 

23.2 

5 

30.5 

29.5 

29.0 

6 

36.6 

35.4 

34.8 

7 

42 7 

41.3 

40.6 

8 

48.8 

47.2 

46.4 

9 

54.9 

53.1 

52.2 


57 

56 

55 

1 

5.7 

5.6 

5.5 

2 

11.4 

11.2 

11.0 

3 

17.1 

16.8 

16.5 

4 

22.8 

22.4 

22.0 

5 

28.5 

28.0 

27.5 

6 

34.2 

33.6 

33.0 

7 

39.9 

39.2 

38.5 

8 

45.6 

44.8 

44.0 

9 

51.3 

50.4 

49.5 


54 

53 

52 

1 

5.4 

5.3 

5.2 

2 

10.8 

10.6 

10.4 

3 

16.2 

15.9 

15.6 

4 

21.6 

21.2 

20.8 

5 

27.0 

26.5 

26.0 

6 

32.4 

31.8 

31.2 

7 

37.8 

37.1 

36.4 

8 

43.2 

42.4 

41.6 

9 

48.6 

47.7 

46.8 


51 

50 

49 


5 

10 

15 

20 

25 

30 

35 

40 

45 



48 

47 

46 


45 

44 

43 

42 


41 

40 

39 


1 

4.8 

4.7 

4.6 

1 

4.5 

4.4 

4.3 

4.2 

1 

4.1 

4 

3.9 

1 

2 

9.6 

9.4 

9.2 

2 

9.0 

8.8 

8.6 

8.4 

2 

8.2 

8 

7.8 

2 

3 

14.4 

14.1 

13.8 

3 

13.5 

13.2 

12.9 

12.6 

3 

12.3 

12 

11.7 

3 

4 

19.2 

18.8 

18.4 

4 

18.0 

17.6 

17.2 

16.8 

4 

16.4 

16 

15.6 

4 

5 

24.0 

23.5 

23.0 

5 

22.5 

22.0 

21.5 

21.0 

5 

20.5 

20 

19.5 

5 

6 

28.8 

28.2 

27.6 

6 

27.0 

26.4 

25.8 

25.2 

6 

24.6 

24 

23.4 

6 

7 

33.6 

32.9 

32.2 

7 

31.5 

30.8 

30.1 

29.4 

7 

28.7 

28 

27.3 

7 

8 

38.4 

37.6 

36.8 

8 

36.0 

35.2 

34.4 

33.6 

8 

32.8 

32 

31.2 

8 

9 

43.2 

42.3 

41.4 

9 

40.5 

39.6 

38.7 

37.8 

9 

36.9 

36 

35.1 

9 


4.9 

9.8 

14.7 

19.6 

24.5 

29.4 

34.3 

39.2 

44.1 


[8] 


Subtract 10 from each entry in this column. 










































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 19° —27°; 63° —71° 


Prop. Parts 



2 

3 

4 

1 

0.2 

0.3 

0.4 

2 

0.4 

0.6 

0.8 

3 

0.6 

0.9 

1.2 

4 

0.8 

1.2 

1.6 

5 

1.0 

1.5 

2.0 

6 

1.2 

1.8 

2.4 

7 

1.4 

2.1 

2.8 

8 

1.6 

2.4 

3.2 

9 

1.8 

2.7 

3.6 



5 

6 

7 

1 

0.5 

0.6 

0.7 

2 

1.0 

1.2 

1.4 

3 

1.5 

1.8 

2.1 

4 

2.0 

2.4 

2.8 

5 

2.5 

3.0 

3.5 

6 

3.0 

3.6 

4.2 

7 

3.5 

4.2 

4.9 

8 

4.0 

4.8 

5.6 

9 

4.5 

5.4 

6.3 



38 

37 

36 

1 

3.8 

3.7 

3.6 

2 

7.6 

7.4 

7.2 

3 

11.4 

11.1 

10.8 

4 

15.2 

14.8 

14.4 

5 

19.0 

18.5 

18.0 

6 

22.8 

22.2 

21.6 

7 

26.6 

25.9 

25.2 

8 

30.4 

29.6 

28.8 

9 

34.2 

33.3 

32.4 



35 

34 

33 

1 

3.5 

3.4 

3.3 

2 

7.0 

6.8 

6.6 

3 

10.5 

10.2 

9.9 

4 

14.0 

13.6 

13.2 

5 

17.5 

17.0 

16.5 

6 

21.0 

20.4 

19.8 

7 

24.5 

23.8 

23.1 

8 

28.0 

27.2 

26.4 

9 

31.5 

30.6 

29.7 


19° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
19 ° 50 ' 
20 ° 00 ' 
10 ' 
20 ' 
30 ' 
40 ' 
20 ° 50 ' 
21 ° 00 ' 
10 ' 
20 ' 
30 ' 
40 ' 
21 ° 50 ' 
22 ° 00 ' 
10 ' 
20 ' 
30 ' 
40 ' 
22 ° 50 ' 
23° 00' 
10 ' 
20 ' 
30 ' 
40 ' 
23 ° 50 ' 
24° 00' 
10 ' 
20 ' 
30 ' 
40 ' 
24 ° 50 ' 
25° 00' 
10 ' 
20 ' 
30 ' 
40 ' 
25 ° 50 ' 
26° 00' 
10 ' 
20 ' 
30 ' 
40 ' 
26 ° 50 ' 
27° 00' 


L Sin* 


9.5126 


.5163 

.5199 

.5235 

.5270 

.5306 

9.5341 

.5375 

.5409 

.5443 

.5477 

.5510 

9.5543 

.5576 

.5609 

.5641 

.5673 

.5704 

9.5736 

.5767 

.5798 

.5828 

.5859 

.5889 

9.5919 

.5948 

.5978 

.6007 

.6036 

.6065 

9.6093 

.6121 

.6149 

.6177 

.6205 

.6232 

9.6259 

.6286 

.6313 

.6340 

.6366 

.6392 

9.6418 

.6444 

.6470 

.6495 

.6521 

.6546 


9.6570 


L Cos* 


37 

36 

36 

35 

36 
35 
34 
34 
34 
34 
33 
33 
33 
33 
32 
32 

31 

32 
31 
31 

30 

31 
30 
30 

29 

30 
29 
29 
29 
28 
28 
28 
28 
28 
27 
27 
27 
27 
27 
26 
26 
26 
26 
26 

25 

26 
25 
24 


L Tan * 


9.5370 


.5411 

.5451 

.5491 

.5531 

.5571 

9.5611 

.5650 

.5689 

.5727 

.5766 

.5804 

9.5842 

.5879 

.5917 

.5954 

.5991 

.6028 

9.6064 

.6100 

.6136 

.6172 

.6208 

.6243 

9.6279 

.6314 

.6348 

.6383 

.6417 

.6452 

9.6486 

.6520 

.6553 

.6587 

.6620 

.6654 

9.6687 

.6720 

.6752 

.6785 

.6817 

.6850 

9.6882 

.6914 

.6946 

.6977 

.7009 

.7040 


9.7072 


c d 


41 

40 

40 

40 

40 

40 

39 

39 

38 

39 
38 
38 

37 

38 
37 
37 
37 
36 
36 
36 
36 
36 

35 

36 
35 

34 

35 

34 

35 
34 
34 

33 

34 

33 

34 
33 
33 

32 

33 

32 

33 
32 
32 
32 

31 

32 

31 

32 


L Cot * c d 


L Cot 

L Cos * 


0.4630 

9.9757 

o 

© 

o 

T—1 

.4589 

.9752 

70 ° 50 ' 

.4549 

.9748 

40 ' 

.4509 

.9743 

30 ' 

.4469 

.9739 

20 ' 

.4429 

.9734 

10 ' 

0.4389 

9.9730 

70° 00' 

.4350 

.9725 

69 ° 50 ' 

.4311 

.9721 

40 ' 

.4273 

.9716 

30 ' 

.4234 

.9711 

20 ' 

.4196 

.9706 

10 ' 

0.4158 

9.9702 

69° 00' 

.4121 

.9697 

68 ° 50 ' 

.4083 

.9692 

40 ' 

4046 

.9687 

30 ' 

.4009 

.9682 

20 ' 

.3972 

.9677 

10 ' 

0.3936 

9.9672 

© 

© 

o 

00 

© 

.3900 

.9667 

67 ° 50 ' 

.3864 

.9661 

40 ' 

.3828 

.9656 

30 ' 

.3792 

.9651 

20 ' 

.3757 

.9646 

10 ' 

0.3721 

9.9640 

67° 00' 

.3686 

.9635 

66 ° 50 ' 

.3652 

.9629 

40 ' 

.3617 

.9624 

30 ' 

.3583 

.9618 

20 ' 

.3548 

.9613 

10 ' 

0.3514 

9.9607 

66° 00' 

.3480 

.9602 

65 ° 50 ' 

.3447 

.9596 

40 ' 

.3413 

.9590 

30 ' 

.3380 

.9584 

20 ' 

.3346 

.9579 

10 ' 

0.3313 

9.9573 

65° 00' 

.3280 

.9567 

64 ° 50 ' 

.3248 

.9561 

40 ' 

.3215 

.9555 

30 ' 

.3183 

.9549 

20 ' 

.3150 

.9543 

10 ' 

0.3118 

9.9537 

64° 00' 

.3086 

.9530 

63 ° 50 ' 

.3054 

.9524 

40 ' 

.3023 

.9518 

30 ' 

.2991 

.9512 

20 ' 

.2960 

.9505 

10 ' 

0.2928 

9.9499 

63° 00' 

L Tan 

L Sin* 

-< - 



32 

31 

30 

29 


28 

27 

26 

25 

24 


1 

3.2 

3.1 

3 

2.9 

1 

2.8 

2.7 

2.6 

2.5 

2.4 

1 

2 

6.4 

6.2 

6 

5.8 

2 

5.6 

5.4 

5.2 

5.0 

4.8 

2 

3 

9.6 

9.3 

9 

8.7 

3 

8.4 

8.1 

7.8 

7.5 

7.2 

3 

4 

12.8 

12.4 

12 

11.6 

4 

11.2 

10.8 

10.4 

10.0 

9.6 

4 

5 

16.0 

15.5 

15 

14.5 

5 

14.0 

13.5 

13.0 

12.5 

12.0 

5 

6 

19.2 

18.6 

18 

17.4 

6 

16.8 

16.2 

15.6 

15.0 

14.4 

6 

7 

22.4 

21.7 

21 

20.3 

7 

19.6 

18.9 

18.2 

17.5 

16.8 

7 

8 

25.6 

24.8 

24 

23.2 

8 

22.4 

21.6 

20.8 

20.0 

19.2 

8 

9 

28.8 

27.9 

27 

26.1 

9 

25.2 

24.3 

23.4 

22.5 

21.6 

9 


[ 9 ] 

























































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 27° —36°; 54° — 63 c 


L Sin* 

9.6570 

.6595 

.6620 

.6644 

.6668 

.6692 

9.6716 

.6740 

.6763 

.6787 

.6810 

.6833 

9.6856 

.6878 

.6901 

.6923 

.6946 

.6968 

9.6990 

.7012 

.7033 

.7055 

.7076 

.7097 

9.7118 

.7139 

.7160 

.7181 

.7201 

.7222 

9.7242 

.7262 

.7282 

.7302 

.7322 

.7342 

9.7361 

.7380 

.7400 

.7419 

.7438 

.7457 

9.7476 

.7494 

.7513 

.7531 

.7550 

.7568 

9.7586 

.7604 

.7622 

.7640 

.7657 

.7675 

9.7692 

L Cos* 


L Cot 

L Cos* 


0.2928 

9.9499 

63° 00' 

.2897 

.9492 

62° 50' 

.2866 

.9486 

40' 

.2835 

.9479 

30' 

.2804 

.9473 

20' 

.2774 

.9466 

10' 

0.2743 

9.9459 

62° 00' 

.2713 

.9453 

61° 50' 

.2683 

.9446 

40' 

.2652 

.9439 

30' 

.2622 

.9432 

20' 

.2592 

.9425 

10' 

0.2562 

9.9418 

61° 00' 

.2533 

.9411 

60° 50' 

.2503 

.9404 

40' 

.2474 

.9397 

30' 

.2444 

.9390 

20' 

.2415 

.9383 

10' 

0.2386 

9.9375 

60° 00' 

.2356 

.9368 

59° 50' 

.2327 

.9361 

40' 

.2299 

.9353 

30' 

.2270 

.9346 

20' 

.2241 

.9338 

10' 

0.2212 

9.9331 

59° 00' 

.2184 

.9323 

58° 50' 

.2155 

.9315 

40' 

.2127 

.9308 

30' 

.2098 

.9300 

20' 

.2070 

.9292 

10' 

0.2042 

9.9284 

58° 00' 

.2014 

.9276 

57° 50' 

.1986 

.9268 

40' 

.1958 

.9260 

30' 

.1930 

.9252 

20' 

.1903 

.9244 

10' 

0.1875 

9.9236 

57° 00' 

.1847 

.9228 

56° 50' 

.1820 

.9219 

40' 

1792 

.9211 

30' 

.1765 

.9203 

20' 

.1737 

.9194 

10' 

0.1710 

9.9186 

56° 00' 

.1683 

.9177 

55° 50' 

.1656 

.9169 

40' 

.1629 

.9160 

30' 

.1602 

.9151 

20' 

.1575 

.9142 

10' 

0.1548 

9.9134 

55° 00' 

.1521 

.9125 

54° 50' 

.1494 

.9116 

40' 

.1467 

.9107 

30' 

.1441 

.9098 

20' 

.1414 

.9089 

10' 

0.1387 

9.9080 

54° 00' 

L Tan 

L Sin* 

- 


27° 00' 

10 ' 
20 ' 
30' 
40' 
27° 50' 

28° 00' 

10 ' 
20 ' 
30' 
40' 
28° 50' 

29° 00' 

10 ' 
20 ' 
30' 
40' 
29° 50' 

30° 00' 

10 ' 
20 ' 
30' 
40' 
30° 50' 

31° 00' 

10 ' 
20 ' 
30' 
40' 
31° 50' 

32° 00' 

10' 
20 ' 
30' 
40' 
32° 50' 

33° 00' 

10 ' 
20 ' 
30' 
40' 
33° 50' 
34° 00' 

10 ' 
20 ' 
30' 
40' 
34° 50' 

35° 00' 

10 ' 
20 ' 
30' 
40' 
35° 50' 
36° 00' 


25 

25 

24 

24 

24 

24 

24 

23 

24 
23 
23 

23 

22 

23 

22 

23 

22 

22 

22 

21 

22 

21 

21 

21 

21 

21 

21 

20 

21 

20 

20 

20 

20 

20 

20 

19 

19 

20 
19 
19 
19 

19 

18 

19 

18 

19 

18 

18 

18 

18 

18 

17 

18 

17 


L Tan* 


9.7072 


c d 


.7103 

.7134 

.7165 

.7196 

.7226 

9.7257 

.7287 

.7317 

.7348 

.7378 

.7408 

9.7438 

.7467 

.7497 

.7526 

.7556 

.7585 

9.7614 

.7644 

.7673 

.7701 

.7730 

.7759 

9.7788 

.7816 

.7845 

.7873 

.7902 

.7930 

9.7958 

.7986 

.8014 

.8042 

.8070 

.8097 

9.8125 

.8153 

.8180 

.8208 

.8235 

.8263 

9.8290 

.8317 

.8344 

.8371 

.8398 

.8425 

9.8452 

.8479 

.8506 

.8533 

.8559 

.8586 


9.8613 


L Cot* 


31 

31 

31 

31 

30 

31 

30 

30 

31 
30 
30 

30 

29 

30 

29 

30 
29 

29 

30 
29 
28 
29 
29 

29 

28 

29 

28 

29 

28 

28 

28 

28 

28 

28 

27 

28 

28 

27 

28 

27 

28 

27 

27 

27 

27 

27 

27 

27 

27 

27 

27 

26 

27 

27 


c d 


Prop. Parts 



31 

30 

29 

1 

3.1 

3 

2.9 

2 

6.2 

6 

5.8 

3 

9.3 

9 

8.7 

4 

12.4 

12 

11.6 

5 

15.5 

15 

14.5 

6 

18.6 

18 

17.4 

7 

21.7 

21 

20.3 

8 

24.8 

24 

23.2 

9 

27.9 

27 

26.1 


28 

27 

26 

1 

2.8 

2.7 

2.6 

2 

5.6 

5.4 

5.2 

3 

8.4 

8.1 

7.8 

4 

11.2 

10.8 

10.4 

5 

14.0 

13.5 

13.0 

6 

16.8 

16.2 

15.6 

7 

19.6 

18.9 

18.2 

8 

22.4 

21.6 

20.8 

9 

25.2 

24.3 

23.4 



25 

24 

1 

2.5 

2.4 

2 

5.0 

4.8 

3 

7.5 

7.2 

4 

10.0 

9.6 

5 

12.5 

12.0 

6 

15.0 

14.4 

7 

17.5 

16.8 

8 

20.0 

19.2 

9 

22.5 

21.6 



23 

22 

1 

2.3 

2.2 

2 

4.6 

4.4 

3 

6.9 

6.6 

4 

9.2 

8.8 

5 

11.5 

11.0 

6 

13.8 

13.2 

7 

16.1 

15.4 

8 

18.4 

17.6 

9 

20.7 

19.8 



21 

20 

1 

2.1 

2 

2 

4.2 

4 

3 

6.3 

6 

4 

8.4 

8 

5 

10.5 

10 

6 

12.6 

12 

7 

14.7 

14 

8 

16.8 

16 

9 

18.9 

18 


[ 10 ] 


* Subtract 10 from each entry in this column. 

































































































































VI. FOUR-PLACE LOGARITHMS OF FUNCTIONS: 36° — 45°• 45° — 54 c 


L Cot 

L Cos * 


0.1387 

9.9080 

54° 00' 

.1361 

.9070 

53 ° 50 ' 

.1334 

.9061 

40 ' 

.1308 

.9052 

30 ' 

.1282 

.9042 

20 ' 

.1255 

.9033 

10 ' 

0.1229 

9.9023 

53° 00' 

.1203 

.9014 

52 ° 50 ' 

.1176 

.9004 

40 ' 

.1150 

.8995 

30 ' 

.1124 

.8985 

20 ' 

.1098 

.8975 

10 ' 

0.1072 

9.8965 

52° 00' 

.1046 

.8955 

51 ° 50 ' 

.1020 

.8945 

40 ' 

.0994 

.8935 

30 ' 

.0968 

.8925 

20 ' 

.0942 

.8915 

10 ' 

0.0916 

9.8905 

51° 00' 

.0890 

.8895 

50 ° 50 ' 

.0865 

.8884 

40 ' 

.0839 

.8874 

30 ' 

.0813 

.8864 

20 ' 

.0788 

.8853 

10 ' 

0.0762 

9.8843 

50° 00' 

.0736 

.8832 

49 ° 50 ' 

.0711 

.8821 

40 ' 

.0685 

.8810 

30 ' 

.0659 

.8800 

20 ' 

.0634 

.8789 

10 ' 

0.0608 

9.8778 

49° 00' 

.0583 

.8767 

48 ° 50 ' 

.0557 

.8756 

40 ' 

.0532 

.8745 

30 ' 

.0506 

.8733 

20 ' 

.0481 

.8722 

10 ' 

0.0456 

9.8711 

00 

o 

O 

© 

.0430 

.8699 

47 ° 50 ' 

.0405 

.8688 

40 ' 

.0379 

.8676 

30 ' 

.0354 

.8665 

20 ' 

.0329 

.8653 

. 10 ' 

0.0303 

9.8641 

<1 

O 

O 

© 

.0278 

.8629 

46 ° 50 ' 

.0253 

.8618 

40 ' 

.0228 

.8606 

30 ' 

.0202 

.8594 

20 ' 

.0177 

.8582 

10 ' 

0.0152 

9.8569 

46° 00' 

.0126 

.8557 

45 ° 50 ' 

.0101 

.8545 

40 ' 

.0076 

.8532 

30 ' 

.0051 

.8520 

20 ' 

.0025 

.8507 

10 ' 

0.0000 

9.8495 

45° 00' 

L Tan 

L Sin* 

-< - 


Prop. Parts 



19 

18 

1 

1.9 

1.8 

2 

3.8 

3.6 

3 

5.7 

5.4 

4 

7.6 

7.2 

5 

9.5 

9.0 

6 

11.4 

10.8 

7 

13.3 

12.6 

8 

15.2 

14.4 

9 

17.1 

16.2 



17 

16 

15 

11 

1.7 

1.6 

1.5 

2 

3.4 

3.2 

3.0 

3 

5.1 

4.8 

4.5 

4 

6.8 

6.4 

6.0 

5 

8.5 

8.0 

7.5 

6 

10.2 

9.6 

9.0 

7 

11.9 

11.2 

10.5 

8 

13.6 

12.8 

12.0 

9 

15.3 

14.4 

13.5 



14 

13 

12 

1 

1.4 

1.3 


1.2 

2 

2.8 

2.6 


2.4 

3 

4.2 

3.9 


3.6 

4 

5.6 

5.2 


4.8 

5 

7.0 

6.5 


6.0 

6 

8.4 

7.8 


7.2 

7 

9.8 

9.1 


8.4 

8 

11.2 

10.4 


9.6 

9 

12.6 

11.7 


10.8 


11 

10 


9 

1 

1.1 

1.0 


0.9 

2 

2.2 

2.0 


1.8 

3 

3.3 

3.0 


2.7 

4 

4.4 

4.0 


3.6 

5 

5.5 

5.0 


4.5 

6 

6.6 

6.0 


5.4 

7 

7.7 

7.0 


6.3 

8 

8.8 

8.0 


7.2 

9 

9.9 

9.0 


8.1 


8 

7 


6 

1 

0.8 

0.7 

0.6 

2 

1.6 

1.4 


1.2 

3 

2.4 

2.1 


1.8 

4 

3.2 

2.8 

2.4 

5 

4.0 

3.5 

3.0 

6 

4.8 

4.2 

3.6 

7 

5.6 

4.9 

4.2 

8 

6.4 

5.6 

4.8 

9 

7.2 

6.3 

5.4 


36° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
36 ° 50 ' 

37° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
37 ° 50 ' 

38° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
38 ° 50 ' 

39° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
39 ° 50 ' 

40° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
40 ° 50 ' 

41° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
41 ° 50 ' 

42° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
42 ° 50 ' 
43° 00' 

10 ' 
20 ' 
30 ' 
40 ' 
43 ° 50 ' 
44° 00' 
10 ' 
20 ' 
30 ' 
40 ' 
44 ° 50 ' 
45° 00' 


L Sin* 


9.7692 


.7710 

.7727 

.7744 

.7761 

.7778 

9.7795 

.7811 

.7828 

.7844 

.7861 

.7877 

9.7893 

.7910 

.7926 

.7941 

.7957 

.7973 

9.7989 

.8004 

.8020 

.8035 

.8050 

.8066 

9.8081 

.8096 

.8111 

.8125 

.8140 

.8155 

9.8169 

.8184 

.8198 

.8213 

.8227 

.8241 

9.8255 

.8269 

.8283 

.8297 

.8311 

.8324 

9.8338 

.8351 

.8365 

.8378 

.8391 

.8405 

9.8418 

.8431 

.8444 

.8457 

.8469 

.8482 


9.8495 


L Cos* 


18 

17 

17 

17 

17 

17 

16 

17 

16 

17 

16 

16 

17 

16 

15 - 

16 

16 

16 

15 

16 
15 

15 

16 

15 

15 

15 

14 

15 
15 

14 

15 

14 

15 
14 
14 

14 

14 

14 

14 

14 

13 

14 

13 

14 
13 

13 

14 

13 

13 

13 

13 

12 

13 

13 


L Tan* 


9.8613 


.8639 

.8666 

.8692 

.8718 

.8745 


9.8771 


.8797 

.8824 

.8850 

.8876 

.8902 


9.8928 


.8954 

.8980 

.9006 

.9032 

.9058 


9.9084 


.9110 

.9135 

.9161 

.9187 

.9212 


9.9238 


.9264 

.9289 

.9315 

.9341 

.9366 


9.9392 


.9417 

.9443 

.9468 

.9494 

.9519 


9.9544 


.9570 

.9595 

.9621 

.9646 

.9671 


9.9697 


.9722 

.9747 

.9772 

.9798 

.9823 


9.9848 


.9874 

.9899 

.9924 

.9949 

.9975 


10.0000 


L Cot* 


c d 


26 

27 

26 

26 

27 

26 

26 

27 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

26 

25 

26 
26 

25 

26 

26 

25 

26 
26 

25 

26 

25 

26 

25 

26 
25 

25 

26 

25 

26 
25 

25 

26 
25 
25 

25 

26 
25 

25 

26 
25 
25 

25 

26 
25 


c d 


* Subtract 10 from each entry in this column. 


[ 11 ] 















































































































VII . FOUR-PLACE VALUES OF FUNCTIONS : 0 ° — 6 °; 84 ° — 90 ‘ 



Sin 

Cos 

Tan 

Cot 

Sec 

Csc 


o 

o 

o 

© 

.0000 

1.000 

.0000 

— 

1.000 

— 

90° 00' 

10 ' 

029 

000 

029 

343.8 

000 

343.8 

89 ° 50 ' 

20 ' 

058 

000 

058 

171.9 

000 

171.9 

40 ' 

30 ' 

.0087 

1.000 

.0087 

114.6 

1.000 

114.6 

30 ' 

40 ' 

116 

.9999 

116 

85.94 

000 

85.95 

20 ' 

0 ° 50 ' 

145 

999 

145 

68.75 

000 

68.76 

10 ' 

1° 00' 

.0175 

.9998 

.0175 

57.29 

l.ooo ” 1 

57.30 

o 

o 

o 

<J* 

oo 

10 ' 

204 

998 

204 

49.10 

000 

49.11 

88 ° 50 ' 

20 ' 

233 

997 

233 

42.96 

000 

42.98 

40 ' 

30 ' 

.0262 

.9997 

.0262 

38.19 

1.000 

38.20 

30 ' 

40 ' 

291 

996 

291 

34.37 

000 

34.38 

20 ' 

1 ° 50 ' 

320 

995 

320 

31.24 

001 

31.26 

10 ' 

o 

o 

o 

C* 

.0349 

.9994 

.0349 

28.64 

1.001 

28.65 

o 

o 

o 

co 

00 

10 ' 

378 

993 

378 

26.43 

001 

26.45 

87 ° 50 ' 

20 ' 

407 

992 

407 

24.54 

001 

24.56 

40 ' 

30 ' 

.0436 

.9990 

.0437 

22.90 

1.001 

22.93 

30 ' 

40 ' 

465 

989 

466 

21.47 

001 

21.49 

20 ' 

2 ° 50 ' 

494 

988 

495 

20.21 

001 

20.23 

10 ' 

3° 00' 

.0523 

.9986 

.0524 

19.08 

1.001 

19.11 

00 

<1 

o 

O 

© 

10 ' 

552 

985 

553 

18.07 

002 

18.10 

86 ° 50 ' 

20 ' 

581 

983 

582 

17.17 

002 

17.20 

40 ' 

30 ' 

.0610 

.9981 

.0612 

16.35 

1.002 

16.38 

30 ' 

40 ' 

640 

980 

641 

15.60 

002 

15.64 

20 ' 

3 ° 50 ' 

669 

978 

670 

14.92 

002 

14.96 

10 ' 

© 

© 

o 

.0698 

.9976 

.0699 

14.30 

1.002 

14.34 

86° 00' 

10 ' 

727 

974 

729 

13.73 

003 

13.76 

85 ° 50 ' 

20 ' 

756 

971 

758 

13.20 

003 

13.23 

40 ' 

30 ' 

.0785 

.9969 

.0787 

12.71 

1.003 

12.75 

30 ' 

40 ' 

814 

967 

816 

12.25 

003 

12.29 

20 ' 

4 ° 50 ' 

843 

964 

846 

11.83 

004 

11.87 

10 ' 

5° 00' 

.0872 

.9962 

.0875 

11.43 

1.004 

11.47 

85° 00' 

10 ' 

901 

959 

904 

11.06 

004 

11.10 

o 

oo 

20 ' 

929 

957 

934 

10.71 

004 

10.76 

40 ' 

30 ' 

.0958 

.9954 

.0963 

10.39 

1.005 

10.43 

30 ' 

40 ' 

.0987 

951 

.0992 

10.08 

005 

10.13 

20 ' 

5 ° 50 ' 

.1016 

948 

.1022 

9.788 

005 

9.839 

10 ' 

6° 00' 

.1045 

.9945 

.1051 

9.514 

1.006 

9.567 

84° 00' 


Cos 

Sin 

Cot 

Tan 

Csc 

Sec 

-c - 


Prop. Parts * 



28 

29 

30 

31 

1 

2.8 

2.9 

3 

3.1 

2 

5.6 

5.8 

6 

6.2 

3 

8.4 

8.7 

9 

9.3 

4 

11.2 

11.6 

12 

12.4 

5 

14.0 

14.5 

15 

15.5 

6 

16.8 

17.4 

18 

18.6 

7 

19.6 

20.3 

21 

21.7 

8 

22.4 

23.2 

24 

24.8 

9 

25.2 

26.1 

27 

27.9 


32 

33 

34 

35 

1 

3.2 

3.3 

3.4 

3.5 

2 

6.4 

6.6 

6.8 

7.0 

3 

9.6 

9.9 

10.2 

10.5 

4 

12.8 

13.2 

13.6 

14.0 

5 

16.0 

16.5 

17.0 

17.5 

6 

19.2 

19.8 

20.4 

21.0 

7 

22.4 

23.1 

23.8 

24.5 

8 

25.6 

26.4 

27.2 

28.0 

9 

28.8 

29.7 

30.6 

31.5 


48 

49 

53 

57 

1 

4.8 

4.9 

5.3 

5.7 

2 

9.6 

9.8 

10.6 

11.4 

3 

14.4 

14.7 

15.9 

17.1 

4 

19.2 

19.6 

21.2 

22.8 

5 

24.0 

24.5 

26.5 

28.5 

6 

28.8 

29.4 

31.8 

34.2 

7 

33.6 

34.3 

37.1 

39.9 

8 

38.4 

39.2 

42.4 

45.6 

9 

43.2 

44.1 

47.7 

51.3 


69 

70 

71 

72 

1 

6.9 

7 

7.1 

7.2 

2 

13.8 

14 

14.2 

14.4 

3 

20.7 

21 

21.3 

21.6 

4 

27.6 

28 

28.4 

28.8 

5 

34.5 

35 

35.5 

36.0 

6 

41.4 

42 

42.6 

43.2 

7 

48.3 

49 

49.7 

50.4 

8 

55.2 

56 

56.8 

57.6 

9 

62.1 

63 

63.9 

64.8 



79 

81 

82 

83 

84 

87 

88 

89 


90 

91 

94 

95 

96 

98 

100 

101 


1 

7.9 

8.1 

8.2 

8.3 

8.4 

8.7 

8.8 

8.9 

1 

9 

9.1 

9.4 

9.5 

9.6 

9.8 

10 

10.1 

1 

2 

15.8 

16.2 

16.4 

16.6 

16.8 

17.4 

17.6 

17.8 

2 

18 

18.2 

18.8 

19.0 

19.2 

19.6 

20 

20.2 

2 

3 

23.7 

24.3 

24.6 

24.9 

25.2 

26.1 

26.4 

26.7 

3 

27 

27.3 

28.2 

28.5 

28.8 

29.4 

30 

30.3 

3 

4 

31.6 

32.4 

32.8 

33.2 

33.6 

34.8 

35.2 

35.6 

4 

36 

36.4 

37.6 

38.0 

38.4 

39.2 

40 

40.4 

4 

5 

39.5 

40.5 

41.0 

41.5 

42.0 

43.5 

44.0 

44.5 

5 

45 

45.5 

47.0 

47.5 

48.0 

49.0 

50 

50.5 

5 

6 

47.4 

48.6 

49.2 

49.8 

50.4 

52.2 

52.8 

53.4 

6 

54 

54.6 

56.4 

57.0 

57.6 

58.8 

60 

60.6 

6 

7 

55.3 

56.7 

57.4 

58.1 

58.8 

60.9 

61.6 

62.3 

7 

63 

63.7 

65.8 

66.5 

67.2 

68.6 

70 

70.7 

7 

8 

63.2 

64.8 

65.6 

66.4 

67.2 

69.6 

70 4 

71.2 

8 

72 

72.8 

75.2 

76.0 

76.8 

78.4 

80 

80.8 

8 

9 

71.1 

72.9 

73.8 

74.7 

75.6 

78.3 

79.2 

80.1 

9 

81 

81.9 

84.6 

85.5 

86.4 

88.2 

90 

90.9 

9 



102 

104 

105 

107 

108 

111 

113 

115 


117 

120 

121 

124 

126 

129 

130 

135 


1 

10.2 

10.4 

10.5 

10.7 

10.8 

11 

11 

12 

1 

12 

12 

12 

12 

13 

13 

13 

14 

1 

2 

20.4 

20.8 

21.0 

21.4 

21.6 

22 

23 

23 

2 

23 

24 

24 

25 

25 

26 

26 

27 

2 

3 

30.6 

31.2 

31.5 

32.1 

32.4 

33 

34 

34 

3 

35 

36 

36 

37 

38 

39 

39 

40 

3 

4 

40.8 

41.6 

42.0 

42.8 

43.2 

44 

45 

46 

4 

47 

48 

48 

50 

50 

52 

52 

54 

4 

5 

51.0 

52.0 

52.5 

53.5 

54.0 

56 

56 

58 

5 

58 

60 

60 

62 

63 

64 

65 

68 

5 

6 

61.2 

62.4 

63.0 

64.2 

64.8 

67 

68 

69 

6 

70 

72 

73 

74 

76 

77 

78 

81 

6 

7 

71.4 

72.8 

73.5 

74.9 

75.6 

78 

79 

80 

7 

82 

84 

85 

87 

88 

90 

91 

94 

7 

8 

81.6 

83.2 

84.0 

85.6 

86.4 

89 

90 

92 

8 

94 

96 

97 

99 

101 

103 

104 

108 

8 

9 

91.8 

93.6 

94.5 

96.3 

97.2 

100 

102 

104 

9 

105 

108 

109 

112 

113 

116 

117 

122 

9 


* If the difference is > 1.00, use Table XI to avoid interpolation. 


[ 12 ] 















































































































VII . FOUR-PLACE VALUES OF FUNCTIONS : 6 ° — 12 °; 78 ° — 84 



[ 13 ] 






























































































































VII . FOUR-PLACE VALUES OF FUNCTIONS : 12 ° — 20 °; 70 ° — 78 ' 



[ 14 ] 



























































































VII. FOUR-PLACE VALUES OF FUNCTIONS: 20° —28°; 62° — 70' 


Prop. Parts 


Sin 

Cos 

Tan 

Cot 

Sec 

Csc 








to 

© 

o 

© 

.3420 

.9397 

.3640 

2.747 

1.064 

2.924 

b 

o 

o 

£ 






10 ' 

448 

387 

673 

723 

065 

901 

69 ° 50 ' 






20 ' 

475 

377 

706 

699 

066 

878 

40 ' 


9 

10 

11 

12 

30 ' 

.3502 

.9367 

.3739 

2.675 

1.068 

2.855 

30 ' 






40 ' 

529 

356 

772 

651 

069 

833 

20 ' 

1 

2 

0.9 

1.8 

1.0 

2 0 

1.1 

2 2 

1.2 

2 4 

20 ° 50 ' 

557 

346 

805 

628 

070 

812 

10 ' 

3 

2.7 

3.0 

3.3 

3.6 

o 

o 

o 

c * 

.3584 

.9336 

.3839 

2.605 

1.071 

2.790 

69° 00' 

4 

5 

3.6 

4.5 

4.0 

5.0 

4.4 

5.5 

4.8 

6 0 

10 ' 

611 

325 

872 

583 

072 

769 

68 ° 50 ' 

6 

5 4 

fi 0 


7 2 

20 ' 

638 

315 

906 

560 

074 

749 

40 ' 

7 

6 3 

7 0 

7 J 7 

8 4 

30 ' 

.3665 

.9304 

.3939 

2.539 

1.075 

2.729 

30 ' 

8 

7 2 

8 0 

8 8 


40 ' 

692 

293 

.3973 

517 

076 

709 

20 ' 

9 

8.1 

9.0 

9^9 

10.8 

21 ° 50 ' 

719 

283 

.4006 

496 

077 

689 

10 ' 






22° 00' 

.3746 

.9272 

.4040 

2.475 

1.079 

2.669 

68° 00' 






10 ' 

773 

261 

074 

455 

080 

650 

67 ° 50 ' 






20 ' 

800 

250 

108 

434 

081 

632 

40 ' 


17 

18 

19 

20 

30 ' 

.3827 

.9239 

.4142 

2.414 

1.082 

2.613 

30 ' 






40 ' 

854 

228 

176 

394 

084 

595 

20 ' 

1 

1.7 

1.8 

1.9 

2 

22 ° 50 ' 

881 

216 

210 

375 

085 

577 

10 ' 

3 

5.1 

5.4 

5.7 

6 

23° 00' 

.3907 

.9205 

.4245 

2.356 

1.086 

2.559 

67° 00' 

4 

6.8 

7.2 

7.6 

8 

10 ' 

934 

194 

279 

337 

088 

542 

66 ° 50 ' 

5 

8.5 

9.0 

9.5 

10 

20 ' 

96’1 

182 

314 

318 

089 

525 

40 ' 

6 

10.2 

10.8 

11.4 

12 

30 ' 

.3987 

.9171 

.4348 

2.300 

1.090 

2.508 

30 ' 

7 

11.9 

12-.6 

13.3 

14 

40 ' 

.4014 

159 

383 

282 

092 

491 

20 ' 

8 

13.6 

14.4 

15.2 

16 

23 ° 50 ' 

041 

147 

417 

264 

093 

475 

10 ' 

9 

15.3 

16.2 

17.1 

18 










24° 00' 

.4067 

.9135 

.4452 

2.246 

1.095 

2.459 

66° 00' 






10 ' 

094 

124 

487 

229 

096 

443 

65 ° 50 ' 






20 ' 

120 

112 

522 

211 

097 

427 * 

40 ' 


25 

26 

27 

28 

30 ' 

.4147 

.9100 

.4557 

2.194 

1.099 

2.411 

30 ' 


40 ' 

173 

088 

592 

177 

100 

396 

20 ' 

1 

2.5 

2.6 

2.7 

2.8 

24 ° 50 ' 

200 

075 

628 

161 

102 

381 

10 ' 

2 

5.0 

5.2 

5.4 

5.6 

25° 00' 

.4226 

.9063 

.4663 

2.145 

1.103 

2.366 

65° 00' 

3 

7.5 

7.8 

8.1 

8.4 









4 

10.0 

10.4 

10.8 

11.2 

10 ' 

253 

051 

699 

128 

105 

352 

64 ° 50 ' 

5 

12.5 

13.0 

13.5 

14.0 

20 ' 

279 

038 

734 

112 

106 

337 

40 ' 

6 

15.0 

15.6 

16.2 

16.8 

30 ' 

.4305 

.9026 

.4770 

2.097 

1.108 

2.323 

30 ' 

7 

17.5 

18.2 

18.9 

19.6 

40 ' 

331 

013 

806 

081 

109 

309 

20 ' 

8 

20.0 

20.8 

21.6 

22.4 

25 ° 50 ' 

358 

.9001 

841 

066 

111 

295 

10 ' 

9 

22.5 

23.4 

24.3 

25.2 

26° 00' 

.4384 

.8988 

' .4877 

2.050 

1.113 

2.281 

o 

o 

o 

<£> 






10 ' 

410 

975 

913 

035 

114 

268 

63 ° 50 ' 






20 ' 

436 

962 

950 

020 

116 

254 

40 ' 






30 ' 

.4462 

.8949 

.4986 

2.006 

1.117 

2.241 

30 ' 


33 

34 

35 

36 

40 ' 

488 

936 

.5022 

1.991 

119 

228 

20 ' 






26 ° 50 ' 

514 

923 

059 

977 

121 

215 

10 ' 

1 

3.3 

3.4 

3.5 

3.6 









2 

6.6 

6.8 

7.0 

7.2 

27° 00' 

.4540 

.8910 

.5095 

1.963 

1.122 

2.203 

63° 00' 

3 

9.9 

10.2 

10.5 

10.8 









4 

13.2 

13.6 

14.0 

14.4 

10 ' 

566 

897 

132 

949 

124 

190 

62 ° 50 ' 

5 

16.5 

17.0 

17.5 

18.0 

20 ' 

592 

884 

169 

935 

126 

178 

40 ' 

6 

19.8 

20.4 

21.0 

21.6 

30 ' 

.4617 

.8870 

.5206 

1.921 

1.127 

2.166 

30 ' 

7 

23.1 

23.8 

24.5 

25.2 

40 ' 

643 

857 

243 

907 

129 

154 

20 ' 

8 

26.4 

27.2 

28.0 

28.8 

27 ° 50 ' 

669 

843 

280 

894 

131 

142 

10 ' 

9 

29.7 

30.6 

31.5 

32.4 

28° 00' 

.4695 

.8829 

.5317 

1.881 

1.133 

2.130 

62° 00' 







Cos 

Sin 

Cot 

Tan 

Csc 

Sec 




52 

54 

55 

56 

57 

58 

59 


60 

62 

63 

64 

65 

67 


1 

5.2 

5.4 

5.5 

5.6 

5.7 

5.8 

5.9 

1 

6 

6.2 

6.3 

6.4 

6.5 

6.7 

1 

2 

10.4 

10.8 

11.0 

11.2 

11.4 

11.6 

11.8 

2 

12 

12.4 

12.6 

12.8 

13.0 

13.4 

2 

3 

15.6 

16.2 

16.5 

16.8 

17.1 

17.4 

17.7 

3 

18 

18.6 

18.9 

19.2 

19.5 

20.1 

3 

4 

20.8 

21.6 

22.0 

22.4 

22.8 

23.2 

23.6 

4 

24 

24.8 

25.2 

25.6 

26.0 

26.8 

4 

5 

26.0 

27.0 

27.5 

28.0 

28.5 

29.0 

29.5 

5 

30 

31.0 

31.5 

32.0 

32.5 

33.5 

5 

6 

31.2 

32.4 

33.0 

33.6 

34.2 

34.8 

35.4 

6 

36 

37.2 

37.8 

38.4 

39.0 

40.2 

6 

7 

36.4 

37.8 

38.5 

39.2 

39.9 

40.6 

41.3 

7 

42 

43.4 

44.1 

44.8 

45.5 

46.9 

7 

8 

41.6 

43.2 

44.0 

44.8 

45.6 

46.4 

47.2 

8 

48 

49.6 

50.4 

51.2 

52.0 

53.6 

8 

9 

46.8 

48.6 

49.5 

50.4 

51.3 

52.2 

53.1 

9 

54 

55.8 

56.7 

57.6 

58.5 

60.3 

9 


[ 15 ] 


















































































VII. FOUR-PLACE VALUES OF FUNCTIONS: 28° —36°; 54° — 62' 



Sin 

Cos 

Tan 

Cot 

Sec 

Csc 



Prop. 

Parts 



28° 00 ' 

.4695 

.8829 

.5317 

1.881 

1.133 

2.130 

62° 00 ' 








10 ' 

720 

816 

354 

868 

134 

118 

61 ° 50 ' 








20 ' 

746 

802 

392 

855 

136 

107 

40 ' 








30 ' 

.4772 

.8788 

.5430 

1.842 

1.138 

2.096 

30 ' 


2 

3 

4 

i 

5 

6 

40 ' 

797 

774 

467 

829 

140 

085 

20 ' 

■ - 







— 

28 ° 50 ' 

823 

760 

505 

816 

142 

074 

10 ' 

1 

0.2 

0.3 

0.4 

0.5 

0.6 









2 

0.4 

0.6 

0.8 

1.0 

1.2 

29° 00' 

.4848 

.8746 

.5543 

1.804 

1.143 

2.063 

61° 00 ' 

3 

0*6 

0.9 

1.2 

1.5 

1.8 

10 ' 

874 

732 

581 

792 

145 

052 

60 ° 50 ' 

4 

5 

0.8 

1.0 

1.2 

1.5 

1.6 

2.0 

2.0 

2.5 

2.4 

3.0 

20 ' 

899 

718 

619 

780 

147 

041 

40 ' 

6 

1.2 

1.8 

2.4 

3.0 

3.6 

30 ' 

.4924 

.8704 

.5658 

1.767 

1.149 

2.031 

30 ' 

7 

1.4 

2.1 

2.8 

3.5 

4.2 

40 ' 

950 

689 

696 

756 

151 

020 

20 ' 

s 

1.6 

2.4 

3.2 

4.0 

4.8 

29 ° 50 ' 

.4975 

675 

735 

744 

153 

010 

10 ' 

9 

1.8 

2.7 

3.6 

4.5 

5.4 

30° 00' 

.5000 

.8660 

.5774 

1.732 

1.155 

2.000 

60° 00 ' 








10 ' 

025 

646 

812 

720 

157 

1.990 

59 ° 50 ' 








20 ' 

050 

631 

851 

709 

159 

980 

40 ' 








30 ' 

.5075 

.8616 

.5890 

1.698 

1.161 

1.970 

30 ' 


ii 

12 

13 

14 

40 ' 

100 

601 

930 

686 

163 

961 

20 ' 








30 ° 50 ' 

125 

587 

.5969 

675 

165 

951 

10 ' 

1 

1.1 

1.2 


1.3 

1.4 









2 

2.2 

2.4 

2.6 

2.8 

31° 00' 

.5150 

.8572 

.6009 

1.664 

1.167 

1.942 

59° 00' 

3 

3.3 

3.6 

3.9 

4.2 

10 ' 

175 

557 

048 

653 

169 

932 

58 ° 50 ' 

4 

R 

4.4 

5.5 

4.8 

6.0 

5.2 

6.5 

5.6 

7.0 

20 ' 

200 

542 

088 

643 

171 

923 

40 ' 

U 

A 

6.6 

7.2 

7.8 

8.4 

30 ' 

.5225 

.8526 

.6128 

1.632 

1.173 

1.914 

30 ' 

o 

7 

7.7 

8 . 4 . 

9.1 

9.8 

40 ' 

250 

511 

168 

621 

175 

905 

20 ' 

g 

8.8 

9.6 

10.4 

11.2 

31 ° 50 ' 

275 

496 

208 

611 

177 

896 

10 ' 

9 

9.9 

10.8 

11.7 

12.6 

32° 00' 

.5299 

.8480 

.6249 

1.600 

1.179 

1.887 

58° 00' 








10 ' 

324 

465 

289 

590 

181 

878 

57 ° 50 ' 








20 ' 

348 

450 

330 

580 

184 

870 

40 ' 








30 ' 

.5373 

.8434 

.6371 

1.570 

1.186 

1.861 

30 ' 


19 

20 

21 

22 

40 ' 

398 

418 

412 

560 

188 

853 

20 ' 








32 ° 50 ' 

422 

403 

453 

550 

190 

844 

10 ' 

1 

1.9 


> 

2.1 

2.2 









2 

3.8 

< 


4.2 

4.4 

33° 00' 

.5446 

.8387 

.6494 

1.540 

1.192 

1.836 

57° 00' 

3 

5.7 

< 


6.3 

6.6 

10 ' 

471 

371 

536 

530 

195 

828 

56 ° 50 ' 

4 

R 

7.6 

8 

10 

8.4 

10.5 

8.8 

11.0 

20 ' 

495 

355 

577 

520 

197 

820 

40 ' 

u 

c 

y.j 

114. 

12 

12.6 

13.2 

30 ' 

.5519 

.8339 

.6619 

1.511 

1.199 

1.812 

3 b ' 

o 

7 

1 l.Tfc 

13.3 

14 

14.7 

15.4 

40 ' 

544 

323 

661 

501 

202 

804 

20 ' 

i 

Q 

1 K O 

16 

16.8 

17.6 

33 ° 50 ' 

568 

307 

703 

492 

204 

796 

10 ' 

o 

9 

17.1 

18 

18^9 

19.8 

34° 00' 

.5592 

.8290 

.6745 

1.483 

' 1.206 

1.788 

56° 00' 








10 ' 

616 

274 

787 

473 

209 

781 

55 ° 50 ' 








20 ' 

640 

258 

830 

464 

211 

773 

40 ' 








30 

.5664 

.8241 

.6873 

1.455 

1.213 

1.766 

30 ' 


37 

38 

39 

40 

40 

688 

225 

916 

446 

216 

758 

20 ' 

— 






— 

34 ° 50 

712 

208 

.6959 

437 

218 

751 

10 ' 

1 

3.7 

*7 A 

3.8 

1 A 

3.9 

7 Q 

4 

Q 

35° 00' 

.5736 

.8192 

.7002 

1.428 

1.221 

1.743 

55° 00' 

2 

3 

7.4 

n.i 

7.0 

11.4 

/ .o 

11.7 

o 

12 

10 ' 

760 

175 

046 

419 

223 

736 

54 ° 50 ' 

4 

14.8 

15.2 

15.6 

16 

20 ' 

783 

158 

089 

411 

226 

729 

40 ' 

5 

18.5 

19.0 

19.5 

20 

30 ' 

.5807 

.8141 

.7133 

1.402 

1.228 

1.722 

30 ' 

6 

22.2 

22.8 

23.4 

24 

40 ' 

831 

124 

177 

393 

231 

715 

20 ' 

7 

25.9 

26.6 

27.3 

28 

35 ° 50 ' 

854 

107 

221 

385 

233 

708 , 

10 ' 

8 

29.6 

30.4 

31.2 

32 









9 

33.3 

S/l 9 

1 

36 

36° 00' 

.5878 

.8090 

.7265 

1.376 

1.236 

1.701 

54° 00' 









Cos 

Sin 

Cot 

Tan 

Csc 

Sec 











44 

45 

46 

47 

48 

49 


50 

51 

52 

53 

54 

55 


i 

4.4 

4.5 

4.6 

4.7 

4.8 

4.9 

1 

5 

5.1 

5.2 

5.3 

5.4 

6.5 

1 

2 

8.8 

9.0 

9.2 

9.4 

9.6 

9.8 

2 

10 

10.2 

10.4 

10.6 

10.8 

11.0 

2 

3 

13.2 

13.5 

13.8 

14.1 

14.4 

14.7 

3 

15 

15.3 

15.6 

15.9 

16.2 

16.5 

3 

4 

17.6 

18.0 

18.4 

18.8 

19.2 

19.6 

4 

20 

20.4 

20.8 

21.2 

21.6 

22.0 

4 

5 

22.0 

22.5 

23.0 

23.5 

24.0 

24.5 

5 

25 

25.5 

26.0 

26.5 

27.0 

27.5 

5 

6 

26.4 

27.0 

27.6 

28.2 

28.8 

29.4 

6 

30 

30.6 

31.2 

31.8 

32.4 

33.0 

6 

7 

30.8 

31.5 

32.2 

32.9 

33.6 

34.3 

7 

35 

35.7 

36.4 

37.1 

37.8 

38.5 

7 

8 

35.2 

36.0 

36.8 

37.6 

38.4 

39.2 

8 

40 

40.8 

41.6 

42.4 

43.2 

44.0 

8 

9 

39.6 

40.5 

41.4 

42.3 

43.2 

44.1 

9 

45 

45.9 

46.8 

47.7 

48.6 

49.5 

9 


[ 16 ] 














































































VII. FOUR-PLACE VALUES OF FUNCTIONS: 36° — 45°; 45° — 54° 







Sin 

Cos 

Tan 

Cot 

Sec 

Csc 


















36° 00' 

.5878 

.8090 

.7265 

1.376 

1.236 

1.701 

54° 00' 


7 

8 

9 


10 

10 ' 

20 ' 

901 

925 

073 

056 

310 

355 

368 

360 

239 

241 

695 

688 

53 ° 50 ' 
40 ' 

1 

2 

3 

0.7 

1.4 

2.1 

0.8 

1.6 

2.4 

0.9 

1.8 

2.7 

1.0 

2.0 

3.0 

30 ' 
40 ' 
36 ° 50 ' 

.5948 

972 

.5995 

.8039 

021 

.8004 

.7400 

445 

490 

1.351 

343 

335 

1.244 

247 

249 

1.681 

675 

668 

30 ' 

20 ' 

10 ' 

5 

3.5 

4.0 

4.5 

5.0 

37° 00' 

.6018 

.7986 

.7536 

1.327 

1.252 

1.662 

53° 00' 

6 

7 

8 

9 

4.2 
4.9 
5.6 

6.3 

4.8 

5.6 

6.4 

7.2 

5.4 

6.3 

7.2 

8.1 

6.0 

7.0 

8.0 

9.0 

10 ' 
20 ' 
30 ' 
40 ' 
37 ° 50 ' 

041 

065 

.6088 

111 

134 

969 

951 

.7934 

916 

898 

581 

627 

.7673 

720 

766 

319 

311 

1.303 

295 

288 

255 

258 

1.260 

263 

266 

655 

649 

1.643 

636 

630 

52 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 





17 


18 

38° db' 

.6157 

.7880 

.7813 

1.280 

1.269 

1.624 

52° 00' 






10 ' 

180 

862 

860 

272 

272 

618 

612 

1.606 

601 

595 

51 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

1 

2 

3 

4 

5 

1.5 
3.0 

4.5 
6.0 

7.5 

1.6 

3.2 

4.8 

6.4 

8 0 

1.7 
3.4 
5.1 

6.8 

8 5 

1.8 

3.6 

5.4 

7.2 

9 n 

20 ' 
30 ' 
40 ' 
38 ° 50 ' 

202 

.6225 

248 

271 

844 

.7826 

808 

790 

907 

.7954 

.8002 

050 

265 

1.257 

250 

242 

275 

1.278 

281 

284 

6 

9.0 

9.6 

11.2 

12.8 

14.4 

10.2 

10.8 

39° 00' 

.6293 

.7771 

.8098 

1.235 

1.287 

1.589 

51° 00' 

8 

9 

10.5 
12.0 

13.5 

11.9 

13.6 

15.3 

12.6 

14.4 

16.2 

10 ' 
20 ' 
30 ' 
40 ' 
39 ° 50 ' 

316 

338 

.6361 

383 

406 

753 

735 

.7716 

698 

679 

146 

195 

.8243 

292 

342 

228 

220 

1.213 

206 

199 

290 

293 

1.296 

299 

302 

583 

578 

1.572 

567 

561 

50 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 


23 

24 

25 


26 

40° 00' 

.6428 

.7660 

.8391 

1.192 

1.305 

1.556 

50° 00 

1 

2 

3 

4 

5 

6 

2.3 

4.6 

6.9 

9.2 

11.5 

13.8 

2.4 

4.8 

7.2 

9.6 

12.0 

14.4 

2.5 
5.0 

7.5 
10.0 
12.5 
15.0 

2.6 

5.2 

7.8 

10.4 

13.0 

15.6 

10 
20 ' 
30 ' 
40 ' 
40 ° 50 ' 

450 

472 

.6494 

517 

539 

642 

623 

.7604 

585 

566 

441 

491 

.8541 

591 

642 

185 

178 

1.171 

164 

157 

309 

312 

1.315 

318 

322 

550 

545 

1.540 

535 

529 

49 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

7 

8 

9 

16.1 

1 O A 

16.8 

mo 

17.5 

oa a 

18.2 

20.8 

23.4 

43 

41° 00' 

.6561 

.7547 

.8693 

1.150 

1.325 

1.524 

49° 00' 


20.7 

41 

21 

.6 i 

* 

22 .J 

12 


10 ' 
20 ' 
30 ' 
40 ' 
41 ° 50 ' 

583 

604 

.6626 

648 

670 

528 

509 

.7490 

470 

451 

744 

796 

.8847 

899 

.8952 

144 

137 

1.130 

124 

117 

328 

332 

1.335 

339 

342 

519 

514 

1.509 

504 

499 

48 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

1 

4.1 

4.2 


4.3 

42° 00' 

.6691 

.7431 

.9004 

1.111 

1.346 

1.494 

48° 00' 

2 

3 

4 

5 

6 

7 

8.2 

12.3 

16.4 

20.5 

24.6 

28.7 

8.4 

12.6 

16.8 

21.0 

25.2 

29.4 


8.6 

12.9 

17.2 

21.5 

25.8 

30.1 

10 ' 
20 ' 
30 ' 
40 ' 
42 ° 50 ' 

713 

734 

.6756 

777 

799 

412 

392 

.7373 

353 

333 

057 

110 

.9163 

217 

271 

104 

098 

1.091 

085 

079 

349 

353 

1.356 

360 

364 

490 

485 

1.480 

476 

471 

47 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

8 

9 

32.8 

36.9 

33.6 

37.8 


34.4 

38.7 

43° 00' 

.6820 

.7314 

.9325 

1.072 

1.367 

1.466 

47° 00' 


56 

57 


58 

10 ' 
20 ' 
30 ' 
40 ' 
43 ° 50 ' 

841 

862 

.6884 

905 

294 

274 

.7254 

234 

380 

435 

.9490 

545 

066 

060 

1.054 

048 

042 

371 

375 

1.379 

382 

386 

462 

457 

1.453 

448 

444 

46 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

1 

5.6 

11.2 

5.7 

11.4 


5.8 

11.6 

926 

214 

601 

2 


44° 00' 

.6947 

.7193 

.9657 

1.036 

1.390 

1.440 

46° 00' 

3 

4 

5 

6 

7 

8 

16.8 

22.4 
28.0 
33.6 
39.2 
44.8 

50.4 

17.1 
22.8 

28.5 

34.2 
39.9 

45.6 

51.3 


17.4 

23.2 

29.0 

34.8 

40.6 

46.4 

52.2 

10 ' 
20 ' 
30 ' 
40 ' 
44 ° 50 ' 

967 

.6988 

.7009 

030 

050 

173 

153 

.7133 

112 

092 

713 

770 

.9827 

884 

.9942 

030 

024 

1.018 

012 

006 

394 

398 

1.402 

406 

410 

435 

431 

1.427 

423 

418 

45 ° 50 ' 
40 ' 
30 ' 
20 ' 
10 ' 

9 


45° 00' 

.7071 

.7071 

1.000 

1.000 

1.414 

1.414 

45° 00' 









r AC 

-|*| 

Cot 

Tan 

Csc 

Sec 










vOS 



_ 


[ 17 ] 
































































VIII. FIVE-PLACE LOGARITHMS: 100—150 


N 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 


Prop 

Parts 

100 

00 000 

043 

087 

130 

173 

217 

260 

303 

346 

389 





01 

432 

475 

518 

561 

604 

647 

689 

732 

775 

817 





02 

00 860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 


44 

43 

42 

03 

01 284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

1 

4.4 

4.3 

4.2 

04 

01 703 

745 

787 

828 

870 

912 

953 

995 

*036 

*078 

2 

3 

8.8 
13 2 

8.6 
12 9 

8.4 

12 6 

05 

02 119 

160 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17.6 

17.2 

16.8 

06 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

5 

22.0 

21.5 

21.0 












6 

26.4 

25.8 

25.2 

07 

02 938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

7 

30.8 

30.1 

29.4 

08 

03 342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

8 

35.2 

34.4 

33.6 

09 

03 743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

9 

39.6 

38.7 

37.8 

110 

04 139 

179 

218 

258 

297 

336 

376 

415 

454 

493 





11 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 


41 

in 

sq 

12 

04 922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 





13 

05 308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

1 

4.1 

4 

3.9 

14 

05 690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

2 

3 

8.2 

12.3 

8 

12 

7.8 

11.7 

15 

06 070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

4 

16.4 

16 

15.6 

16 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 

20.5 

20 

19.5 












6 

24.6 

24 

23.4 

17 

06 819 

856 

893 

930 

967 

*004 

*041 

*078 

*115 

*151 

7 

28.7 

28 

27.3 

18 

07 188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

8 

32.8 

32 

31.2 

19 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

9 

36.9 

36 

35.1 

120 

07 918 

954 

990 

*027 

*063 

*099 

*135 

*171 

*207 

*243 





21 

08 279 

314 

350 

386 

422 

458 

493 

529 

565 

600 


38 

37 

36 

99 

636 

679 

707 

743 

778 

814 

849 

884 

920 

955 





23 

08 991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

1 

3.8 

3.7 

3.6 












2 

7.6 

7.4 

7.2 

24 

09 342 

377 

412 

447 

482 

517 

552 

587 

621 

656 

3 

11.4 

11.1 

10.8 

25 

09 691 

726 

760 

795 

830 

864 

899 

934 

968 

*003 

4 

15.2 

14.8 

14.4 

26 

10 037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

19.0 

18.5 

18.0 












6 

22.8 

22.2 

21.6 

27 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

7 

26.6 

25.9 

25.2 

28 

10 721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

8 

30.4 

29.6 

28.8 

29 

11 059 

093 

126 

160 

193 

227 

261 

294 

327 

361 

9 

34.2 

33.3 

32.4 

130 

394 

428 

461 

494 

528 

561 

594 

628 

661 

694 





31 

11 727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 


35 

34 

33 

32 

12 057 

090 

123 

156 

189 

222 

254 

287 

320 

352 





33 

385 

418 

450 

483 

516 

548 

581 

613 

646 

678 

1 

3.5 

3.4 

3.3 












2 

7.0 

6.8 

6.6 

34 

12 710 

743 

775 

808 

840 

872 

905 

937 

969 

*001 

3 

10.5 

10.2 

9.9 

35 

13 033 

066 

098 

130 

162 

194 

226 

258 

290 

322 

4 

14.0 

13.6 

17 n 

13.2 

1 (K £ 

36 

354 

386 

418 

450 

481 

513 

545 

577 

609 

640 

0 

6 

1 / .0 

21.0 

1 / .u 

20.4 

10.0 

19.8 

37 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

7 

8 

24.5 
28 0 

23.8 
27 2 

23.1 

26 4 

38 

13 988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

9 

31.5 

30.6 

29.7 

39 

14 301 

333 

364 

395 

426 

457 

489 

520 

551 

582 





140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 





41 

14 922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 


32 

31 

30 

42 

15 229 

259 

290 

320 

351 

381 

412 

442 

473 

503 

1 

3 2 

3.1 

3 

43 

534 

564 

594 

625 

655 

685 

715 

746 

776 

806 

2 

6.4 

6.2 

6 

44 

15 836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

3 

4 

9.6 

12.8 

9.3 

12.4 

9 

12 

45 

16 137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

5 

16.0 

15.5 

15 

46 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

6 

19.2 

18.6 

18 












7 

22.4 

21.7 

21 

47 

16 732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

8 

25.6 

24.8 

24 

48 

17 026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

9 

28.8 

27.9 

27 

49 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 





150 

17 609 

638 

667 

696 

725 

754 

782 

811 

840 

869 





N 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 


Prop 

Parts 


[ 18 ] 






































































































VIII. FIVE-PLACE LOGARITHMS: 150 — 200 


Prop . Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 





150 

17 609 

638 

667 

696 

725 

754 

782 

811 

840 

869 





51 

17 898 

926 

955 

984 

*013 

*041 

*070 

*099 

*127 

*156 




52 

18 184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

l 

2.9 

2.8 

53 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

2 

3 

5.8 

8.7 

5.6 

8 4 

54 

18 752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 

4 

11.6 

11.2 

55 

19 033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

5 

14.5 

14.0 

56 

312 

340 

368 

396 

424 

451 

479 

507 

535 

562 

6 

7 

17.4 

20.3 

16.8 

19.6 

57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

8 

23.2 

22.4 

58 

19 866 

893 

921 

948 

976 

*003 

*030 

*058 

*085 

*112 

9 

26.1 

25.2 

59 

20 140 

167 

194 

222 

249 

276 

303 

330 

358 

385 





160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 


9.7 

26 

61 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 





62 

20 952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

1 

2.7 

2.6 

63 

21 219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

3 

5.4 

8.1 

5.2 

7.8 

64 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

4 

10.8 

10.4 

65 

21 748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

5 

13.5 

13.0 

66 

22 011 

037 

063 

089 

115 

141 

167 

194 

220 

246 

6 

16.2 

15.6 












7 

18.9 

18.2 

67 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

8 

21.6 

20.8 

68 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

9 

24.3 

23.4 

69 

22 789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 





170 

23 045 

070 

096 

121 

147 

172 

198 

223 

249 

274 




71 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 




72 

553 

578 

603 

629 

654 

679 

704 

729 . 

754 

779 


1 

2.5 

73 

23 805 

830 

855 

880 

905 

930 

955 

980 

*005 

*030 


2 

5.0 













3 

7.5 

74 

24 055 

080 

105 

130 

155 

180 

204 

229 

254 

279 


4 

10.0 

75 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 


5 

12.5 

76 

551 

576 

601 

625 

650 

674 

699 

724 

748 

773 


6 

15.0 













7 

17.5 

77 

24 797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 


8 

20.0 

78 

25 042 

066 

091 

115 

139 

164 

188 

212 

237 

261 


9 

22.5 

79 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 





180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 


24 

23 

81 

25 768 

792 

816 

840 

864 

888 

912 

935 

959 

983 





82 

26 007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 

2.4 

2.3 

83 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

2 

4.8 

4.6 












3 

7.2 

6.9 

84 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

4 

9.6 

9.2 

85 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

5 

12.0 

11.5 

86 

26 951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

6 

14.4 

13.8 












7 

16.8 

16.1 

87 

27 184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

8 

19.2 

18.4 

88 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

9 

21.6 

20.7 

89 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 





190 

27 875 

898 

921 

944 

967 

989 

*012 

*035 

*058 

*081 


22 

21 

91 

28 103 

126 

149 

171 

194 

217 

240 

262 

285 

307 





92 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

1 

2.2 

2.1 

93 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2 












3 

6.6 

6.3 

94 

28 780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

4 

8 

.8 

8.4 

95 

29 003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

5 

6 

11.0 

13.2 

10.5 

12.6 

96 

226 

248 

270 

292 

314 

336 

358 

380 

403 

425 

7 

Q 

15.4 

14.7 

1 A Q 

97 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

O 

9 

17.6 
io a 

lo.o 

18.9 

98 

667 

688 

710 

732 

754 

776 

798 

820 

842 - 

863 





99 

29 885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 





200 

30 103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

Prop . Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 19 ] 













































VIII. FIVE-PLACE LOGARITHMS: 200 — 250 


N 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 

200 

30 103 

125 

146 

168 

190 

211 

233 

255 

276 

298 




01 

320 

341 

363 

384 

406 

428 

449 

471 

492 

514 




02 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 


22 

21 

03 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

— 















1 


2.2 

2.1 

04 

30 963 

984 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

2 


4.4 

4.2 

05 

31 175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

3 


6.6 

6.3 

06 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

4 


8.8 

8.4 












5 

11.0 

10.5 

07 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

6 

13.2 

12.6 

08 

31 806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

7 

15.4 

14.7 

09 

32 015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

o 

9 

17.0 

19.8 

lo.o 

18.9 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

408 




11 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 




12 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 


20 

13 

32 838 

858 

879 

899 

919 

940 

960 

980 

*001 

*021 

1 

2 

14 

33 041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

2 

4 

15 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

3 

A 

6 

O 

16 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

% 

5 

O 

10 

17 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

6 

12 

18 

33 846 

866 

885 

905 

925 

945 

965 

985 

*005 

*025 

1 

8 

14 

16 

19 

34 044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

9 

18 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 




21 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 




22 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 


19 

23 

34 830 

850 

869 

889 

908 

928 

947 

967 

986 

*005 

1 

1.9 

24 

35 025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

2 

o 

3.8 

k 7 

25 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

4 

O. 4 

7 6 

26 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

5 

9.5 

27 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

6 

7 

11.4 

13 3 

28 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

8 

15.2 

29 

35 984 

*003 

*021 

*040 

*059 

*078 

*097 

*116 

*135 

*154 

9 

17.1 

230 

36 173 

192 

211 

229 

248 

267 

286 

305 

324 

342 




31 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 



32 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 


lo 

33 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

1 

1.8 

34 

36 922 

940 

959 

977 

996 

*014 

*033 

*051 

*070 

*088 

2 

3 

3.6 

5.4 

35 

37 107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

4 

7.2 

36 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

5 

9.0 












6 

10.8 

37 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

7 

12.6 

38 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

8 

14.4 

39 

37 840 

858 

876 

894 

912 

931 

949 

967 

985 

*003 

9 

16.2 

240 

38 021 

039 

057 

075 

093 

112 

130 

148 

166 

184 




41 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 


17 


42 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 





43 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

1 

1.7 












2 

3.4 

44 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

3 

5.1 

45 

38 917 

934 

952 

970 

987 

*005 

*023 

*041 

*058 

*076 

4 

6.8 

46 

39 094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

5 

6 

8.5 

10.2 

47 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

7 

Q 

11.9 

1 O ft 

48 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

O 

9 

10.0 

1 A 3 

49 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 




250 

39 794 

811 

829 

846 

863 

881 

898 

915 

933 

950 




N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

Parts 


[ 20 ] 














































































































































VIII. FIVE-PLACE LOGARITHMS: 250 — 300 


Prop. 

Parts 


18 

1 

1.8 

2 

3.6 

3 

5.4 

4 

7.2 

5 

9.0 

6 

10.8 

7 

12.6 

8 

14.4 

9 

16.2 


17 

1 

1.7 

2 

3.4 

3 

5.1 

4 

6.8 

5 

8.5 

6 

10.2 

7 

11.9 

8 

13.6 

9 

15.3 


16 

1 

1.6 

2 

3.2 

3 

4.8 

4 

6.4 

5 

8.0 

6 

9.6 

7 

11.2 

8 

12.8 

9 

14.4 


15 

1 

1.5 

2 

3.0 

3 

4.5 

4 

6.0 

5 

7.5 

6 

9.0 

7 

10.5 

8 

12.0 

9 

13.5 


14 

1 

1.4 

2 

2.8 

3 

4.2 

4 

5.6 

5 

7.0 

6 

8.4 

7 

9.8 

8 

11.2 

9 

12.6 

Prop. Parts 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

250 

39 794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

51 

39 967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 

52 

40 140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

53 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

54 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

55 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

56 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

57 

40 993 

*010 

*027 

*044 

*061 

*078 

*095 

*m 

*128 

*145 

58 

41 162 

179 

196 

212 

229 

246 

263 

280 

296 

313 

59 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 

61 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

62 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

63 

41 996 

*012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

*144 

64 

42 160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

65 

325 

341 

357 

374 

390 

406 

423 

439 

455 

472 

66 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

67 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

68 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

69 

42 975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

270 

43 136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

71 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

72 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

73 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

74 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

75 

43 933 

949 

965 

981 

996 

*012 

*028 

*044 

*059 

*075 

76 

44 091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

77 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

78 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

79 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

81 

44 871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 

82 

45 025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

83 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

84 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

85 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

86 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

87 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

88 

45 939 

954 

969 

984 

*000 

*015 

*030 

*045 

*060 

*075 

89 

46 090 

105 

120 

135 

150 

165 

180 

195 

210 

225 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

91 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

92 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

93 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

94 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

95 

46 982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

*114 

96 

47 129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

97 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

98 

422 

436 

451 

465 

480 

494 

509 

524 

538 

553 

99 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

300 

47 712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 21 ] 












































































































































VIII. FIVE-PLACE LOGARITHMS: 300 — 350 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 

300 

47 712 

727 

741 

756 

770 

784 

799 

813 

828 

842 



01 

47 857 

871 

885 

900 

914 

929 

943 

958 

972 

986 



02 

48 001 

015 

029 

044 

058 

073 

087 

101 

116 

130 



03 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 



04 

287 

302 

316 

330 

344 

359 

373 

387 

401 

416 


15 

05 

430 

444 

458 

473 

487 

501 

515 

530 

544 

558 

1 

1.5 

06 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

2 

3.0 












3 

4.5 

07 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

4 

6.0 

08 

855 

869 

883 

897 

911 

926 

940 

954 

968 

982 

5 

7.5 

09 

48 996 

*010 

*024 

*038 

*052 

*066 

*080 

*094 

*108 

*122 

6 

9.0 

10.5 












7 

310 

49 136 

150 

164 

178 

192 

206 

220 

234 

248 

262 

8 

12.0 

11 

276 

290 

304 

318 

332 

346 

360 

374 

388 

402 

9 

13.5 

12 

415 

429 

443 

457 

471 

485 

499 

513 

527 

541 



13 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 



14 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 



15 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 


14 

16 

49 969 

982 

996 

*010 

*024 

*037 

*051 

*065 

*079 

*092 


17 

50 106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

1 

1.4 

18 

243 

*256 

270 

284 

297 

311 

325 

338 

352 

365 

2 

3 

4 

2.8 

4.2 

5.6 

19 

379 

393 

406 

420 

433 

447 

461 

474 

488 

501 

320 

515 

529 

542 

556 

569 

583 

596 

610 

623 

637 

5 

7.0 












A 

8.4 













21 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

7 

9.8 

22 

786 

799 

813 

826 

840 

853 

866 

880 

893 

907 

8 

11.2 

23 

50 920 

934 

947 

961 

974 

987 

*001 

*014 

*028 

*041 

9 

12.6 

24 

51 055 

068 

081 

095 

108 

121 

135 

148 

162 

175 



25 

188 

202 

215 

228 

242 

255 

268 

282 

295 

308 



26 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 



27 

455 

468 

481 

495 

508 

521 

534 

548 

561 

574 


13 

28 

29 

587 

601 

614 

627 

640 

654 

667 

680 

812 

693 

706 

838 


720 

733 

746 

759 

772 

786 

799 

825 

1 

1.3 

330 

851 

865 

878 

891 

904 

917 

930 

943 

957 

970 

2 

3 

2.6 

3.9 

31 

51 983 

996 

*009 

*022 

*035 

*048 

*061 

*075 

*088 

*101 

4 

5.2 

32 

52 114 

127 

140 

153 

166 

179 

192 

205 

218 

231 

0 

$ 

6.5 

7.8 

9.1 

33 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

7 

34 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

8 

9 

10.4 

11.7 

35 

504 

517 

530 

543 

556 

569 

582 

595 

608 

621 


36 

634 

647 

660 

673 

686 

699 

711 

724 

737 

750 



37 

763 

776 

789 

802 

815 

827 

840 

853 

866 

879 



38 

52 892 

905 

917 

930 

943 

956 

969 

982 

994 

*007 



39 

53 020 

033 

046 

058 

071 

084 

097 

110 

122 

135 


12 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 



1 

1.2 

41 

275 

288 

301 

314 

326 

339 

352 

364 

377 

390 

2 

2.4 

42 

403 

415 

428 

441 

453 

466 

479 

491 

504 

517 

3 

3.6 

43 

529 

542 

555 

567 

580 

593 

605 

618 

631 

643 

4 

4.8 









5 

6.0 

44 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

6 

7.2 

45 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

7 

3 

8.4 
q 6 

46 

53 908 

920 

933 

945 

958 

970 

983 

995 

*008 

*020 

9 

10.8 

47 

54 033 

045 

058 

070 

083 

095 

108 

120 

133 

145 



48 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 



49 

283 

295 

307 

320 

332 

345 

357 

370 

382 

394 



350 

54 407 

419 

432 

444 

456 

469 

481 

494 

506 

518 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 


[ 22 ] 





































































































































VIII. FIVE-PLACE LOGARITHMS: 350 — 400 


Prop 

. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



350 

54 407 

419 

432 

444 

456 

469 

481 

494 

506 

518 



51 

531 

543 

555 

568 

580 

593 

605 

617 

630 

642 



52 

654 

667 

679 

691 

704 

716 

728 

741 

753 

765 



53 

777 

790 

802 

814 

827 

839 

851 

864 

876 

888 


13 

54 

54 900 

913 

925 

937 

949 

962 

974 

986 

998 

*011 

1 

1.3 

55 

55 023 

035 

047 

060 

072 

084 

096 

108 

121 

133 

2 

2.6 

56 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

3 

4 

3.9 

5.2 

57 

267 

279 

291 

303 

315 

328 

340 

352 

364 

376 

5 

6.5 

58 

388 

400 

413 

425 

437 

449 

461 

473 

485 

497 

6 

7.8 

59 

509 

522 

534 

546 

558 

570 

582 

594 

606 

618 

7 

8 

9.1 

10.4 

360 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

9 

11.7 

61 

751 

763 

775 

787 

799 

811 

823 

835 

847 

859 



62 

871 

883 

895 

907 

919 

931 

943 

955 

967 

979 



63 

55 991 

*003 

*015 

*027 

*038 

*050 

*062 

*074 

*086 

*098 



64 

56 110 

122 

134 

146 

158 

170 

182 

194 

205 

217 



65 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 


12 

66 

348 

360 

372 

384 

396 

407 

419 

431 

443 

455 

1 

1.2 

67 

467 

478 

490 

502 

514 

526 

538 

549 

561 

573 

2 

2.4 

68 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

3.6 

69 

703 

714 

726 

738 

750 

761 

773 

785 

797 

808 

A 

A 8 












5 

6.0 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

6 

7 

7.2 

8 4 

71 

56 937 

949 

961 

972 

984 

996 

*008 

*019 

*031 

*043 

8 

9.6 

72 

57 054 

066 

078 

089 

101 

113 

124 

136 

148 

159 

9 

10.8 

73 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 



74 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 



75 

403 

415 

426 

438 

449 

461 

473 

484 

496 

507 



76 

519 

530 

542 

553 

565 

576 

588 

600 

611 

623 



77 

634 

646 

657 

669 

680 

692 

703 

715 

726 

738 


11 

78 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

1 

1.1 

79 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 

3 

2.2 

3 3 

380 

57 978 

990 

*001 

*013 

*024 

*035 

*047 

*058 

*070 

*081 

4 

4.4 

81 

58 092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

5 

5.5 

82 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

6 

7 

6.6 

7.7 

83 

320 

331 

343 

354 

365 

377 

388 

399 

410 

422 

8 

8.8 

84 

433 

444 

456 

467 

478 

490 

501 

512 

524 

535 

9 

9.9 

85 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 



86 

659 

670 

681 

692 

704 

715 

726 

737 

749 

760 



87 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 



88 

883 

894 

906 

917 

928 

939 

950 

961 

973 

984 


10 

89 

58 995 

*006 

*017 

*028 

*040 

*051 

*062 

*073 

*084 

*095 

— 

390 

59 106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

1 

2 

1.0 

2 0 

91 

218 

229 

240 

251 

262 

273 

284 

295 

306 

318 

3 

3.0 

92 

329 

340 

351 

362 

373 

384 

395 

406 

417 

428 

4 

4.0 

93 

439 

450 

461 

472 

483 

494 

506 

517 

528 

539 

5 

6 

5.0 

6.0 

94 

550 

561 

572 

583 

594 

605 

616 

627 

638 

649 

7 

7.0 

95 

660 

671 

682 

693 

704 

715 

726 

737 

748 

759 

8 

8.0 

96 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

9 

9.0 














97 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 



98 

59 988 

999 

*010 

*021 

*032 

*043 

*054 

*065 

*076 

*086 



99 

60 097 

108 

119 

130 

141 

152 

163 

173 

184 

195 



400 

60 206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 23 ] 







































































































VIII. FIVE-PLACE LOGARITHMS: 400 — 450 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 

400 

60 206 

217 

228 

239 

249 

260 

271 

282 

293 

304 



01 

314 

325 

336 

347 

358 

369 

379 

390 

401 

412 



02 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 



03 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 



04 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 



05 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 



06 

853 

863 

874 

885 

895 

■ 906 

917 

927 

938 

949 



07 

60 959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 


11 

08 

61 066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

1 

1.1 

09 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 

2.2 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

3 

4 

3.3 

4.4 

11 

384 

395 

405. 

416 

426 

437 

448 

458 

469 

479 

5 

5.5 

12 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

6 

7 

6.6 

7 7 

13 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

8 

8^ 

14 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

9 

9.9 

15 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 



16 

61 909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 



17 

62 014 

024 

034 

045 

055 

066 

076 

086 

097 

107 



18 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 



19 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 



420 

325 

335 

346 

356 

366 

377 

387 

397 

408 

418 



21 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 


10 

22 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 



23 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

1 

1.0 












2 

2.0 

24 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

3.0 

25 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

4 

4.0 

26 

62 941 

951 

961 

972 

982 

992 

*002 

*012 

*022 

*033 

5 

5.0 












6 

6.0 

27 

63 043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

7 

7.0 

28 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

8 

8.0 

29 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

9 

9.0 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 



31 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 



32 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 



33 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 



34 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 



35 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 



36 

63 949 

959 

969 

979 

988 

998 

*008 

*018 

*028 

*038 


9 

37 

64 048 

058 

068 

078 

088 

098 

108 

118 

128 

137 


ft ft 

38 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

1 

2 

u.y 

18 

39 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

3 

2.7 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

4 

5 

3.6 

4.5 

41 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

6 

5.4 

42 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

7 

6.3 

43 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

8 

9 

7.2 

8.1 

44 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 



45 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 



46 

64 933 

943 

953 

963 

972 

982 

992 

*002 : 

*011 

*021 



47 

65 031 

040 

050 

060 

070 

079 

089 

099 

108 

118 



48 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 



49 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 



450 

65 321 

331 

341 

350 

360 

369 

379 

389 

398 

408 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 


[ 24 ] 























































VIII. FIVE-PLACE LOGARITHMS: 450 — 500 


Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



450 

65 321 

331 

341 

350 

360 

369 

379 

389 

398 

408 



51 

418 

427 

437 

447 

456 

466 

475 

485 

495 

504 



52 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 



53 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 



54 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 



55 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 



56 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 


10 

57 

65 992 

*001 

*011 

*020 

*030 

*039 

*049 

*058 

*068 

*077 

1 

1.0 

58 

66 087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

2 

2.0 

59 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 














3 

0 .1) 












4 

4.0 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

361 

5 

6 

5.0 

6 0 

61 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

7 

7.0 

62 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

8 

8.0 

63 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

9 

9.0 

64 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 



65 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 



66 

839 

848 

857 

867 ’ 

876 

885 

894 

904 

913 

922 



67 

66 932 

941 

950 

960 

969 

978 

987 

997 

*006 

*015 



68 

67 025 

034 

043 

052 

062 

071 

080 

089 

099 

108 



69 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 



470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 


9 

71 

302 

311 

321 

330 

339 

348 

357 

367 

376 

385 



72 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

1 

0.9 

73 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

2 

1.8 












3 

2.7 

74 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

4 

3.6 

75 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

5 

4.5 

76 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

6 

5.4 












7 

6.3 

77 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

8 

7.2 

78 

67 943 

952 

961 

970 

979 

988 

997 

*006 

*015 

*024 

9 

8.1 

79 

68 034 

043 

052 

061 

070 

079 

088 

097 

106 

115 



480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 



81 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 



82 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 



83 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 



84 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 



85 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 


8 

86 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

i 

0 8 

87 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

2 

1 6 

88 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

3 

2.4 

89 

68 931 

940 

949 

958 

966 

975 

984 

993 

*002 

*011 

4 

5 

3.2 

4.0 

490 

69 020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

6 

4.8 

91 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

7 

5.6 

92 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

8 

9 

6.4 

7.2 

93 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 



94 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 



95 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 



96 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 



97 

636 

644 

653 

662 

671 

679 

688 

697 

705 

714 



98 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 



99 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 



500 

69 897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 25 ] 






































































































































VIII. FIVE-PLACE LOGARITHMS: 500 — 550 


N 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Prop 

. Parts 

500 

69 897 

906 

914 

923 

932 

940 

949 

958 

966 

975 



01 

69 984 

992 

*001 

*010 

*018 

*027 

*036 

*044 

*053 

*062 



02 

70 070 

079 

088 

096 

105 

114 

122 

131 

140 

148 



03 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 



04 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 



05 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 



06 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 



07 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 


9 

08 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

1 

0.9 

09 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

2 

1.8 

510 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

3 

4 

2.7 

3.6 

11 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

5 

c 

4.5 

c A 

12 

70 927 

935 

944 

952 

961 

969 

978 

986 

995 

*003 

D 

7 

0.3: 

6 3 

13 

71 012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

8 

7.2 

14 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

9 

8.1 

15 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 



16 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 



17 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 



18 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 



19 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 



520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 



21 

684 

692 

700 

709 

717 

725 

734 

742 

750 

759 


8 

22 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 



23 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

1 

0.8 












2 

1.6 

24 

71 933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

3 

2.4 

25 

72 016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

4 

3.2 

26 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4.0 












6 

4.8 

27 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

7 

5.6 

28 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

8 

6.4 

29 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

9 

7.2 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 



31 

509 

518 

526 

534 

542 

550 

558 

567 

575 

583 



32 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 



33 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 



34 

754 

762 

770 

779 

787 

795 

803 

811 

819 

827 



35 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 



36 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 


7 

37 

72 997 

*006 

*014 

*022 

*030 

*038 

*046 

*054 

*062 

*070 



38 

73 078 

086 

094 

102 

111 

119 

127 

135 

143 

151 

1 

o 

0.7 

1 4 

39 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

3 

2.1 

540 

239 

247 

255 

263 

272 

280 

288 

296 

304 

312 

4 

2.8 













3 5 

41 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

6 

4.2 

42 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

7 

4.9 

43 

480 

488 

496 

504 

512 

520 

528 

536 

544 

552 

8 

5.6 












9 

6.3 

44 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 



45 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 



46 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 



47 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 



48 

878 

886 

894 

902 

910 

918 

926 

933 

941 

949 



49 

73 957 

965 

973 

981 

989 

997 

*005 

*013 

*020 

*028 



550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

Parts 


[ 26 ] 























































































































VIII. FIVE-PLACE LOGARITHMS: 550 — 600 


Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



550 

74 036 

044 

052 

060 

068 

076 

084 

092 

099 

107 



51 

115 

123 

131 

139 

147 

155 

162 

170 

178 

186 



52 

194 

202 

210 

218 

225 

233 

241 

249 

257 

265 



53 

273 

280 

288 

296 

304 

312 

320 

327 

335 

343 



54 

351 

359 

367 

374 

382 

390 

398 

406 

414 

421 



55 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 



56 

507 

515 

523 

531 

539 

547 

554 

562 

570 

578 



57 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 



58 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 



59 

741 

749 

757 

764 

772 

780 

788 

796 

803 

811 



560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 



61 

896 

904 

912 

920 

927 

935 

943 

950 

958 

966 


8 

62 

74 974 

981 

989 

997 

*005 

*012 

*020 

*028 

*035 

*043 

1 

0.8 

63 

75 051 

059 

066 

074 

082 

089 

097 

105 

113 

120 

2 

3 

1.6 

2 4 

64 

128 

136 

143 

151 

159 

166 

174 

182 

189 

197 

4 

3.2 

65 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

5 

4.0 

66 

282 

289 

297 

305 

312 

320 

328 

335 

343 

351 

6 

7 

4.8 

5.6 

67 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

8 

6.4 

68 

435 

442 

450 

458 

465 

473 

481 

488 

496 

504 

9 

7.2 

69 

511 

519 

526 

534 

542 

549 

557 

565 

572 

580 



570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 



71 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 



72 

740 

747 

755 

762 

770 

778 

785 

793 

800 

808 



73 

815 

823 

831 

838 

846 

853 

861 

868 

876 

884 



74 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 



75 

75 967 

974 

982 

989 

997 

*005 

*012 

*020 

*027 

*035 



76 

76 042 

050 

057 

065 

072 

080 

087 

095 

103 

110 



77 

118 

125 

133 

140 

148 

155 

163 

170 

178 

185 



78 

193 

200 

208 

215 

223 

230 

238 

245 

253 

260 



79 

268 

275 

283 

290 

298 

305 

313 

320 

328 

335 



580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 


7 

81 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 


f 

82 

492 

500 

507 

515 

522 

530 

537 

545 

552 

559 

1 

0.7 

83 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

2 

1.4 












3 

2.1 

84 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

4 

2.8 

85 

716 

723 

730 

738 

745 

753 

760 

768 

775 

782 

5 

3.5 

86 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

6 

4.2 












7 

4.9 

87 

864 

871 

879 

886 

893 

901 

908 

916 

923 

930 

8 

5.6 

88 

76 938 

945 

953 

960 

967 

975 

982 

989 

997 

*004 

9 

6.3 

89 

77 012 

019 

026 

034 

041 

048 

056 

063 

070 

078 



590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 



91 

159 

166 

173 

181 

188 

195 

203 

210 

217 

225 



92 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 



93 

305 

313 

320 

327 

335 

342 

349 

357 

364 

371 



94 

379 

386 

393 

401 

408 

415 

422 

430 

437 

444 



95 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 



96 

525 

532 

539 

546 

554 

561 

568 

576 

583 

590 



97 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 



98 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 



99 

743 

750 

757 

764 

772 

779 

786 

793 

801 

808 



600 

77 815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

Prop. 

Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 27 ] 

















































































































VIII. FIVE-PLACE LOGARITHMS: 600 — 650 


N 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

Parts 

600 

77 815 

822 

830 

837 

844 

851 

859 

866 

873 

880 



01 

887 

895 

902 

909 

916 

924 

931 

938 

945 

952 



02 

77 960 

967 

974 

981 

988 

996 

*003 

*010 

*017 

*025 



03 

78 032 

039 

046 

053 

061 

068 

075 

082 

089 

097 



04 

104 

111 

118 

125 

132 

140 

147 

154 

161 

168 



05 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 



06 

247 

254 

262 

269 

276 

283 

290 

297 

305 

312 


Q 

07 

319 

326 

333 

340 

347 

355 

362 

369 

376 

383 

— 

O 

08 

390 

398 

405 

412 

419 

426 

433 

440 

447 

455 

1 

0.8 

09 

462 

469 

476 

483 

490 

497 

504 

512 

519 

526 

2 

3 

1.6 

2 4 

610 

533 

540 

547 

554 

561 

569 

576 

583 

590 

597 

4 

3.2 

11 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

5 

6 

4.0 

4.8 

12 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

7 

5.6 

13 

746 

753 

760 

767 

774 

781 

789 

796 

803 

810 

8 

6.4 












9 

7.2 

14 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 



15 

888 

895 

902 

909 

916 

923 

930 

937 

944 

951 



16 

78 958 

965 

972 

979 

986 

993 

*000 

*007 

*014 

*021 



17 

79 029 

036 

043 

050 

057 

964 

071 

078 

085 

092 



18 

099 

106 

113 

120 

127 

134 

141 

148 

155 

162 



19 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 



620 

239 

246 

253 

260 

267 

274 

281 

288 

295 

302 



21 

309 

316 

323 

330 

337 

344 

351 

358 

365 

372 


7 

22 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 



23 

449 

456 

463 

470 

477 

484 

491 

498 

505 

511 

1 

2 

0.7 

1.4 

24 

518 

525 

532 

539 

546 

553 

560 

567 

574 

581 

3 

2.1 

25 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

4 

2.8 

26 

657 

664 

671 

678 

685 

692 

699 

706 

713 

720 

5 

6 

3.5 

4.2 

27 

727 

734 

741 

748 

754 

761 

768 

775 

782 

789 

7 

4.9 

28 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

8 

Q 

5.6 | 

a ^ 

29 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 


0.0 

630 

79 934 

941 

948 

955 

962 

969 

975 

982 

989 

996 



31 

80 003 

010 

017 

024 

030 

037 

044 

051 

058 

065 



32 

072 

079 

085 

092 

099 

106 

113 

120 

127 

134 



33 

140 

147 

154 

161 

168 

175 

182 

188 

195 

202 



34 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 



35 

277 

284 

291 

298 

305 

312 

318 

325 

332 

339 



36 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 


6 

37 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

1 

0.6 

38 

482 

489 

496 

502 

509 

516 

523 

530 

536 

543 

2 

1.2 

39 

550 

557 

564 

570 

577 

584 

591 

598 

604 

611 

3 

1.8 












4 

z.4 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

5 

3.0 

41 

686 

693 

699 

706 

713 

720 

726 

733 

740 

747 

6 

7 

3.6 

4 9 

42 

754 

760 

767 

774 

781 

787 

794 

801 

808 

814 

8 

4.8 

43 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

9 

5.4 

44 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 



45 

80 956 

963 

969 

976 

983 

990 

996 

*003 

*010 

*017 



46 

81 023 

030 

037 

043 

050 

057 

064 

070 

077 

084 



47 

090 

097 

104 

111 

117 

124 

131 

137 

144 

151 



48 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 



49 

224 

231 

238 

245 

251 

258 

265 

271 

278 

285 



650 

81 291 

298 

305 

311 

318 

325 

331 

338 

345 

351 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 


[ 28 ] 














































































































VIII. FIVE-PLACE LOGARITHMS: 650 — 700 


Prop 

Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



650 

81 291 

298 

305 

311 

318 

325 

331 

338 

345 

351 



51 

358 

365 

371 

378 

385 

391 

398 

405 

411 

418 



52 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 



53 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 



54 

558 

564 

571 

578 

584 

591 

598 

604 

611 

617 



55 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 



56 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 



57 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 



58 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 



59 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 



660 

81 954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 



61 

82 020 

027 

033 

040 

046 

053 

060 

066 

073 

079 


7 

62 

086 

092 

099 

105 

112 

119 

125 

132 

138 

145 

1 

0.7 

63 

151 

158 

164 

171 

178 

184 

191 

197 

204 

210 

2 

1.4 

64 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

4 

2 ft 

65 

282 

289 

295 

302 

308 

315 

321 

328 

334 

341 

5 

3.5 

66 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

6 

7 

4.2 

4 Q 

67 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

8 

5.6 

68 

478 

484 

491 

497 

504 

510 

517 

523 

530 

536 

9 

6.3 

69 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 



670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 



71 

672 

679 

685 

692 

698 

705 

711 

718 

724 

730 



72 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 



73 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 



74 

866 

872 

879 

885 

892 

898 

905 

911 

918 

924 



75 

930 

937 

943 

950 

956 

963 

969 

975 

982 

988 



76 

82 995 

*001 

*008 

*014 

*020 

*027 

*033 

*040 

*046 

*052 



77 

83 059 

065 

072 

078 

085 

091 

097 

104 

110 

117 



78 

123 

129 

136 

142 

149 

155 

161 

168 

174 

181 



79 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 



680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 



81 

315 

321 

327 

334 

340 

347 

353 

359 

366 

372 


b 

82 

378 

385 

391 

398 

404 

410 

417 

423 

429 

436 

1 

0.6 

83 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

2 

3 

1.2 

1.8 

84 

506 

512 

518 

525 

531 

537 

544 

550 

556 

563 

4 

2.4 

85 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

5 

3.0 

86 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

6 

3.6 












7 

4.2 

87 

696 

702 

708 

715 

721 

727 

734 

740 

746 

753 

8 

4.8 

88 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

9 

5.4 

89 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 



690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 



91 

83 948 

954 

960 

967 

973 

979 

985 

992 

998 

*004 



92 

84 011 

017 

023 

029 

036 

042 

048 

055 

061 

067 



93 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 



94 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 



95 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 



96 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 



97 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 



98 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 



99 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 



700 

84 510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

Prop. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 29 ] 










































VIII. FIVE-PLACE LOGARITHMS: 700 — 750 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop 

i. Parts 

700 

84 510 

516 

522 

528 

535 

541 

547 

553 

559 

566 



01 

572 

578 

584 

590 

597 

603 

609 

615 

621 

628 



02 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 



03 

696 

702 

708 

714 

720 

726 

733 

739 

745 

751 



04 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 



05 

819 

825 

831 

837 

844 

850 

856 

862 

868 

874 



06 

880 

887 

893 

899 

905 

911 

917 

924 

930 

936 



07 

84 942 

948 

954 

960 

967 

973 

979 

985 

991 

997 


7 

08 

85 003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

1 

0.7 

09 

065 

071 

077 

083 

089 

095 

101 

107 

114 

120 

2 

1.4 

710 

126 

132 

138 

144 

150 

156 

163 

169 

175 

181 

3 

4 

2.1 

2.8 

11 

187 

193 

199 

205 

211 

217 

224 

230 

236 

242 

5 

3.5 

12 

248 

254 

260 

266 

272 

278 

285 

291 

297 

303 

6 

7 

4.2 

4.9 

13 

309 

315 

321 

327 

333 

339 

345 

352 

358 

364 

8 

5.6 

14 

370 

376 

382 

388 

394 

400 

406 

412 

418 

425 

9 

6.3 

15 

431 

437 

443 

449 

455 

461 

467 

473 

479 

485 



16 

491 

497 

503 

509 

516 

522 

528 

534 

540 

546 



17 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 



18 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 



19 

673 

679 

685 

691 

697 

703 

709 

715 

721 

727 



720 

733 

739 

745 

751 

757 

763 

769 

775 

781 

788 



21 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 


ft 

22 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 


O 

23 

914 

920 

926 

932 

938 

944 

950 

956 

962 

968 

1 

0.6 

24 

85 974 

980 

986 

992 

998 

*004 

*010 

*016 

*022 

*028 

2 

3 

1.2 

1.8 

25 

86 034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

4 

2.4 

26 

094 

100 

106 

112 

118 

124 

130 

136 

141 

147 

6 

3.0 












6 

3.6 

27 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

7 

4.2 

28 

213 

219 

225 

231 

237 

243 

249 

255 

261 

267 

8 

4.8 

29 

273 

279 

285 

291 

297 

303 

308 

314 

320 

326 

9 

5.4 

730 

332 

338 

344 

350 

356 

362 

368 

374 

380 

386 



31 

392 

398 

404 

410 

415 

421 

427 

433 

439 

445 



32 

451 

457 

463 

469 

475 

481 

487 

493 

499 

504 



33 

510 

516 

522 

528 

534 

540 

546 

552 

558 

564 



34 

570 

576 

581 

587 

593 

599 

605 

611 

617 

623 



35 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 



36 

688 

694 

700 

705 

711 

717 

723 

729 

735 

741 


5 

37 

747 

753 

759 

764 

770 

776 

782 

788 

794 

800 



38 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

1 

A 

0.5 

39 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

L 

3 

1.0 

1.5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

4 

2.0 












K 

2.5 












u 

41 

86 982 

988 

994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

6 

3.0 

42 

87 040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

7 

3.5 

43 

099 

105 

111 

116 

122 

128 

134 

140 

146 

151 

8 

4.0 












9 

4.5 

44 

157 

163 

169 

175 

181 

186 

192 

198 

204 

210 



45 

216 

221 

227 

233 

239 

245 

251 

256 

262 

268 



46 

274 

280 

286 

291 

297 

303 

309 

315 

320 

326 



47 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 



48 

390 

396 

402 

408 

413 

419 

425 

431 

437 

442 



49 

448 

454 

460 

466 

471 

477 

483 

489 

495 

500 



750 

87 506 

512 

518 

523 

529 

535 

541 

547 

552 

558 



N 

0 

1 

2 

3 1 

4 

5 

6 

7 

8 

9 

Prop. 

Parts 


[ 30 ] 



































VIII. FIVE-PLACE LOGARITHMS: 750 — 800 


Prop 

Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



750 

87 506 

512 

518 

523 

529 

535 

541 

547 

552 

558 



51 

564 

570 

576 

581 

587 

593 

599 

604 

610 

616 



52 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 



53 

679 

685 

691 

697 

703 

708 

714 

720 

726 

731 



54 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 



55 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 



56 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 



57 

910 

915 

921 

927 

933 

938 

944 

950 

955 

961 



58 

87 967 

973 

978 

984 

990 

996 

*001 

*007 

*013 

*018 



59 

88 024 

030 

036 

041 

047 

053 

058 

064 

070 

076 



760 

081 

087 

093 

098 

104 

110 

116 

121 

127 

133 



61 

138 

144 

150 

156 

161 

167 

173 

178 

184 

190 


6 

62 

195 

201 

207 

213 

218 

224 

230 

235 

241 

247 

1 

0.6 

63 

252 

258 

264 

270 

275 

281 

287 

292 

298 

304 

2 

3 

1.2 

1 ft 

64 

309 

315 

321 

326 

332 

338 

343 

349 

355 

360 

4 

2.4 

65 

366 

372 

377 

383 

389 

395 

400 

406 

412 

417 

5 

3.0 

66 

423 

429 

434 

440 

446 

451 

457 

463 

468 

474 

6 

7 

3.6 

4.2 

67 

480 

485 

491 

497 

502 

508 

513 

519 

525 

530 

8 

4.8 

68 

536 

542 

547 

553 

559 

564 

570 

576 

581 

587 

9 

6.4 

69 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 



770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 



71 

705 

711 

717 

722 

728 

734 

739 

745 

750 

756 



72 

762 

767 

773 

779 

784 

790 

795 

801 

807 

812 



73 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 



74 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 



75 

930 

936 

941 

947 

953 

958 

964 

969 

975 

981 



76 

88 986 

992 

997 

*003 

*009 

*014 

*020 

*025 

*031 

*037 



77 

89 042 

048 

053 

059 

064 

070 

076 

081 

087 

092 



78 

098 

104 

109 

115 

120 

126 

131 

137 

143 

148 



79 

154 

159 

165 

170 

176 

182 

187 

193 

198 

204 



780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 


5 

81 

265 

271 

276 

282 

287 

293 

298 

304 

310 

315 



82 

321 

326 

332 

337 

343 

348 

354 

360 

365 

371 

1 

0.6 

83 • 

376 

382 

387 

393 

398 

404 

409 

415 

421 

426 

2 

1.0 












3 

1.5 

84 

432 

437 

443 

448 

454 

459 

465 

470 

476 

481 

4 

2.0 

85 

487 

492 

498 

504 

509 

515 

520 

526 

531 

537 

5 

2.5 ^ 

86 

542 

548 

553 

559 

564 

570 

575 

581 

586 

592 

6 

3.0 












7 

3.5 

87 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

8 

4.0 

88 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

9 

4.5 

89 

708 

713 

719 

724 

730 

735 

741 

746 

752 

757 



790 

763 

768 

774 

779 

785 

790 

796 

801 

807 

812 



91 

818 

823 

829 

834 

840 

845 

851 

856 

862 

867 



92 

873 

878 

883 

889 

894 

900 

905 

911 

916 

922 



93 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 



94 

89 982 

988 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 



95 

90 037 

042 

048 

053 

059 

064 

069 

075 

080 

086 



96 

091 

097 

102 

108 

113 

119 

124 

129 

135 

140 



97 

146 

151 

157 

162 

168 

173 

179 

184 

189 

195 



98 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 



99 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 



800 

90 309 

314 

320 

325 

331 

336 

342 

347 

352 

358 

Prop 

. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 31 ] 

































VIII. FIVE-PLACE LOGARITHMS: 800 — 850 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

Parts 

800 

90 309 

314 

320 

325 

331 

336 

342 

347 

352 

358 



01 

363 

369 

374 

380 

385 

390 

396 

401 

407 

412 



02 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 



03 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 



04 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 



05 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 



06 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 



07 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 



08 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 



09 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 



810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 



11 

902 

907 

913 

918 

924 

929 

934 

940 

945 

950 



12 

90 956 

961 

966 

972 

977 

982 

988 

993 

998 

*004 


6 

13 

91 009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

1 

0.6 

14 

062 

068 

073 

078 

084 

089 

094 

100 

105 

110 

2 

3 

1.2 

1 8 

15 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

4 

2.4 

16 

169 

174 

180 

185 

190 

196 

201 

206 

212 

217 

5 

3.0 

17 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6 

7 

3.6 

4.2 

18 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

8 

4.8 

19 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 

9 

5.4 

820 

381 

387 

392 

397 

403 

408 

413 

418 

424 

429 



21 

434 

440 

445 

450 

455 

461 

466 

471 

477 

482 



22 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 



23 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 



24 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 



25 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 



26 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 



27 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 



28 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 



29 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 



830 

908 

913 

918 

924 

929 

934 

939 

944 

950 

955 



31 

91 960 

965 

971 

976 

981 

986 

991 

997 

*002 

*007 


5 

32 

92 012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

— 


33 

065 

070 

075 

080 

085 

091 

096 

101 

106 

111 

1 

0.5 












2 

1.0 

34 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

3 

1.5 

35 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

4 

2.0 

36 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

2.5 












6 

3.0 

37 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

7 

3.5 

38 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

8 

9 

4.0 

4 5 

39 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 



840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 



41 

480 

485 

490 

495 

500 

505 

511 

516 

521 

526 



42 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 



43 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 



44 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 



45 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 



46 

737 

742 

747 

752 

758 

763 

768 

773 

778 

783 



47 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 



48 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 



49 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 



850 

92 942 

947 

952 

957 

962 

967 

973 

978 

983 

988 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 


[ 32 ] 
























































































































VIII. FIVE-PLACE LOGARITHMS: 850 — 900 


Prop 

. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



850 

92 942 

947 

952 

957 

962 

967 

973 

978 

983 

988 



51 

92 993 

998 

*003 

*008 

*013 

*018 

*024 

*029 

*034 

*039 



52 

93 044 

049 

054 

059 

064 

069 

075 

080 

085 

090 



53 

095 

100 

105 

110 

115 

120 

125 

131 

136 

141 



54 

146 

151 

156 

161 

166 

171 

176 

181 

186 

192 



55 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 



56 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 


6 

57 

298 

303 

308 

313 

318 ' 

323 

328 

334 

339 

344 

1 

0.6 

58 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

2 

1.2 

59 

399 

404 

409 

414 

420 

425 

430 

435 

440 

445 

3 

4 

1.8 

2.4 

860 

450 

455 

460 

465 

470 

475 

480 

485 

490 

495 

5 

3.0 

61 

500 

505 

510 

515 

520 

526 

531 

536 

541 

546 

7 


62 

551 

556 

561 

566 

571 

576 

581 

586 

591 

596 

8 

4.8 

63 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

9 

5.4 

64 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 



65 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 



66 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 



67 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 



68 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 



69 

902 

907 

912 

917 

922 

927 

932 

937 

942 

947 



870 

93 952 

957 

962 

967 

972 

977 

982 

987 

992 

997 



71 

94 002 

007 

012 

017 

022 

027 

032 

037 

042 

047 



72 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

1 

0.5 

73 

101 

106 

111 

116 

121 

126 

131 

136 

141 

146 

2 

3 

1.0 

1.5 

74 

151 

156 

161 

166 

171 

176 

181 

186 

191 

196 

4 

2.0 

75 

201 

206 

211 

216 

221 

226 

231 

236 

240 

245 

5 

2.5 

76 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

6 

3.0 












7 

3.5 

77 

300 

305 

310 

315 

320 

325 

330 

335 

340 

345 

8 

4.0 

78 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

9 

4.5 

79 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 



880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 



81 

498 

503 

507 

512 

517 

522 

527 

532 

537 

542 



82 

547 

552 

557 

562 

567 

571 

576 

581 

586 

591 



83 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 



84 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 



85 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 


A. 

86 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

— 


87 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

1 

0.4 

88 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

2 

3 

0.8 

1.2 

89 

890 

895 

900 

905 

910 

915 

919 

924 

929 

934 

4 

1.6 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

5 

6 

2.0 

2.4 

91 

94 988 

993 

998 

*002 

*007 

*012 

*017 

*022 

*027 

*032 

7 

2.8 

92 

95 036 

041 

046 

051 

056 

061 

066 

071 

075 

080 

8 

3.2 

93 

085 

090 

095 

100 

105 

109 

114 

119 

124 

129 

9 

3.6 














94 

134 

139 

143 

148 

153 

158 

163 

168 

173 

177 



95 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 



96 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 



97 

279 

284 

289 

294 

299 

303 

308 

313 

318 

323 



98 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 



99 

376 

381 

386 

390 

395 

400 

405 

410 

415 

419 



900 

95 424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

Prop. 

Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 33 ] 











































VIII. FIVE-PLACE LOGARITHMS: 900 — 950 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. Parts 

900 

95 424 

429 

434 

439 

444 

448 

453 

458 

463 

468 



01 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 



02 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 



03 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 



04 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 



05 

665 

670 

674 

679 

684 

689 

694 

698 

703 

708 



06 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 



07 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 



08 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 



09 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 



910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

947 



11 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 


K 

12 

95 999 

*004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 



13 

96 047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

1 

0.5 

14 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

2 

3 

1.0 

1.5 

15 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

4 

2.0 

16 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

2.5 












6 

3.0 

17 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

7 

3.5 

18 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

8 

4.0 

19 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

9 

4.5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

417 

421 



21 

426 

431 

435 

440 

445 

450 

454 

459 

464 

468 



22 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 



23 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 



24 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 



25 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 



26 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 



27 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 



28 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 



29 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 



930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 



31 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 


4 

32 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

— 


33 

96 988 

993 

997 

*002 

*007 

*011 

*016 

*021 

*025 

*030 

1 

0.4 












2 

0.8 

34 

97 035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

3 

1.2 

35 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

4 

1.6 

36 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

6 

2.0 

2.4 

37 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

7 

2.8 

38 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

o 

Q 

o.Z 

3 6 

39 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 



940 

313 

317 

322 

327 

331 

336 

340 

345 

350 

354 



41 

359 

364 

368 

373 

377 

382 

387 

391 

396 

400 



42 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 



43 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 



44 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 



45 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 



46 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 



47 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 



48 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 



49 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 



950 

97 772 

777 

782 

786 

791 

795 

800 

804 

809 

813 



N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop 

. Parts 


[ 34 ] 


























































































































VIII. FIVE-PLACE LOGARITHMS: 950 — 1000 


Prop 

. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 



950 

97 772 

777 

782 

786 

791 

795 

800 

804 

809 

813 



51 

818 

823 

827 

832 

836 

841 

845 

850 

855 

859 



52 

864 

868 

873 

877 

882 

886 

891 

896 

900 

905 



53 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 



54 

97 955 

959 

964 

968 

973 

978 

982 

987 

991 

996 



55 

98 000 

005 

009 

014 

019 

023 

028 

032 

037 

041 



56 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 



57 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 



58 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 



59 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 



960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 



61 

272 

277 

281 

286 

290 

295 

299 

304 

308 

313 



62 

318 

322 

327 

331 

336 

340 

345 

349 

354 

358 

1 

0.5 

63 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

2 

3 

1.0 

1.5 

64 

408 

412 

417 

421 

426 

430 

435 

439 

444 

448 

4 

2.0 

65 

453 

457 

462 

466 

471 

475 

480 

484 

489 

493 

5 

2.5 

66 

498 

502 

507 

511 

516 

520 

525 

529 

534 

538 

6 

7 

3.0 

3.5 

67 

543 

547 

552 

556 

561 

565 

570 

574 

579 

583 

8 

4.0 

68 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

9 

4.5 

69 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 



970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 



71 

722 

726 

731 

735 

740 

744 

749 

753 

758 

762 



72 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 



73 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 



74 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 



75 

900 

905 

909 

914 

918 

923 

927 

932 

936 

941 



76 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 



77 

98 989 

994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 



78 

99 034 

038 

043 

047 

052 

056 

061 

065 

069 

074 



79 

078 

083 

087 

092 

096 

100 

105 

109 

114 

118 



980 

123 

127 

131 

136 

140 

145 

149 

154 

158 

162 


4 

81 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 



82 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

1 

0.4 

83 

255 

260 

264 

269 

273 

277 

282 

286 

291 

295 

2 

0.8 












3 

1.2 

84 

300 

304 

308 

313 

317 

322 

326 

330 

335 

339 

4 

1.6 

85 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

5 

2.0 

86 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

6 

2.4 












7 

2.8 

87 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

8 

A 

3.2 

Q ft 

88 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

y 

0.0 

89 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 



990 

564 

568 

572 

577 

581 

585 

590 

594 

599 

603 



91 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 



92 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 



93 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 



94 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 



95 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 



96 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 



97 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 



98 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 



99 

99 957 

961 

965 

970 

974 

978 

983 

987 

991 

996 



1000 

00 000 

004 

009 

013 

017 

022 

026 

030 

035 

039 

Prop 

. Parts 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 35 ] 





































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS— 0 C 


/ 

L Sin * 

d 

L Tan * 

c d 

L Cot 

L Cos * 


o 






10.00 000 

60 

1 

6.46 

373 


6.46 

373 


3.53 

627 

000 

59 

2 

6.76 

476 


6.76 

476 


3.23 

524 

000 

58 

3 

6.94 

085 


6.94 

085 


3.05 

915 

000 

57 

4 

7.06 

579 


7.06 

579 


2.93 

421 

000 

56 

5 

7.16 

270 


7.16 

270 


2.83 

730 

10.00 000 

55 

6 

7.24 

188 


7.24 

188 


2.75 

812 

000 

54 

7 

7.30 

882 


7.30 

882 


2.69 

118 

000 

53 

8 

7.36 

682 


7.36 

682 


2.63 

318 

000 

52 

9 

7.41 

797 


7.41 

797 


2.58 

203 

000 

51 

10 

7.46 

373 


7.46 

373 


2.53 

627 

10.00 000 

50 

11 

7.50 

512 


7.50 

512 


2.49 

488 

000 

49' 

12 

7.54 

291 


7.54 

291 


2.45 

709 

000 

48 

13 

7.57 

767 


7.57 

767 


2.42 

233 

000 

47 

14 

7.60 

985 


7.60 

986 


2.39 

014 

000 

46 

15 

7.63 

982 


7.63 

982 


2.36 

018 

10.00 000 

45 

16 

7.66 

784 


7.66 

785 


2.33 

215 

10.00 000 

44 

17 

7.69 

417 


7.69 

418 


2.30 

582 

9.99 999 

43 

18 

7.71 

900 


7.71 

900 


2.28 

100 

999 

42 

19 

7.74 

248 


7.74 

248 


2.25 

752 

999 

41 

20 

7.76 

475 


7.76 

476 


2.23 

524 

9.99 999 

40 

21 

7.78 

594 


7.78 

595 


2.21 

405 

999 

39 

22 

7.80 

615 


7.80 

615 


2.19 

385 

999 

38 

23 

7.82 

545 


7.82 

546 


2.17 

454 

999 

37 

24 

7.84 

393 


7.84 

394 


2.15 

606 

999 

36 

25 

7.86 

166 


7.86 

167 


2.13 

833 

9.99 999 

35 

26 

7.87 

870 


7.87 

871 


2.12 

129 

999 

34 

27 

7.89 

509 


7.89 

510 


2.10 

490 

999 

33 

28 

7.91 

088 


7.91 

089 


2.08 

911 

999 

32 

29 

7.92 

612 


7.92 

613 


2.07 

387 

998 

31 

30 

7.94 

084 


7.94 

086 


2.05 

914 

9.99 998 

30 

31 

7.95 

508 


7.95 

510 


2.04 

490 

998 

29 

32 

7.96 

887 


7.96 

889 


2.03 

111 

998 

28 

33 

7.98 

223 


7.98 

225 


2.01 

775 

998 

27 

34 

7.99 

520 


7.99 

522 


2.00 

478 

998 

26 

35 

8.00 

779 


8.00 

781 


1.99 

219 

9.99 998 

25 

36 

8.02 

002 


8.02 

004 


1.97 

996 

998 

24 

37 

8.03 

192 


8.03 

194 


1.96 

806 

997 

23 

38 

8.04 

350 


8.04 

353 


1.95 

647 

997 

22 

39 

8.05 

478 


8.05 

481 


1.94 

519 

997 

21 

40 

8.06 

578 


8.06 

581 


1.93 

419 

9.99 997 

20 

41 

8.07 

650 


8.07 

653 


1.92 

347 

997 

19 

42 

8.08 

696 


8.08 

700 


1.91 

300 

997 

18 

43 

8.09 

718 


8.09 

722 


1.90 

278 

997 

17 

44 

8.10 

717 

999 

8.10 

720 

998 

1.89 

280 

996 

16 

45 

8.11 

693 

976 

8.11 

696 

976 

1.88 

304 

9.99 996 

15 

46 

8.12 

647 

954 

8.12 

651 

955 

1.87 

349 

996 

14 

47 

8.13 

581 

934 

8.13 

585 

934 

1.86 

415 

996 

13 

48 

8.14 

495 

914 

8.14 

500 

915 

1.85 

500 

996 

12 

49 

8.15 

391 

896 

8.15 

395 

895 

1.84 

605 

996 

11 

50 

8.16 

268 

877 

8.16 

273 

878 

1.83 

727 

9.99 995 

10 

51 

8.17 

128 

860 

8.17 

133 

860 

1.82 

867 

995 

9 

52 

8.17 

971 

o4o 

8.17 

976 

o4o 

1.82 

024 

995 

8 

53 

8.18 

798 

oZ i 

8.18 

804 

ozo 

1.81 

196 

995 

7 

54 

8.19 

610 

olz 

8.19 

616 

olz 

1.80 

384 

995 

6 

55 

8.20 

407 

/y / 

8.20 

413 

/ y / 

1.79 

587 

9.99 994 

5 

56 

8.21 

189 

782 

8.21 

195 

782 

7SKC\ 

1.78 

805 

994 

4 

57 

8.21 

958 

i oy 

8.21 

964 


1.78 

036 

994 

3 

58 

8.22 

713 

/ oo 

7/IQ 

8.22 

720 

7 AO 

1.77 

280 

994 

2 

59 

8.23 

456 

i *±o 

7QH 

8.23 

462 

4 

1.76 

538 

994 

1 

60 

8.24 

186 

/ oU 

8.24 

192 


1.75 

808 

9.99 993 

0 


L Cos * 

d 

L Cot * 

c d 

L Tan 

L Sin * 

/ 


Prop. Parts 


Avoid inaccurate interpo¬ 
lation by using Table X on 
page 83. 

Subtract 10 from each 
entry in the columns marked 
with “ * 99 in the table. 


717 


72 

143 

215 

287 

358 

430 

502 

574 

645 

663 


66 

133 

199 

265 

332 

398 

464 

530 

597 


616 


62 

123 

185 

246 

308 

370 

431 

493 

554 


576 


58 

115 

172 

230 

288 

345 

402 

460 

518 


540 


54 

108 

162 

216 

270 

324 

378 

432 

486 


508 


51 

102 

152 

203 

254 

305 

356 

406 

457 


480 


48 

96 

144 

192 

240 

288 

336 

384 

432 


706 


71 

141 

212 

282 

353 

424 

494 

565 

635 


653 


65 

130 

196 

261 

326 

392 

457 

522 

588 


608 


61 

122 

182 

243 

304 

365 

426 

486 

547 


568 


57 

114 

170 

227 

284 

341 

398 

454 

511 


533 


53 

107 

160 

213 

266 

320 

373 

426 

480 


502 


50 

100 

151 

201 

251 

301 

351 

402 

452 


475 


48 

95 

142 

190 

238 

285 

332 

380 

428 


695 


70 

139 

208 

278 

348 

417 

486 

556 

626 


643 


64 

129 

193 

257 

322 

386 

450 

514 

579 


599 


560 


56 

112 

168 

224 

280 

336 

392 

448 

504 


526 


53 

105 

158 

210 

263 

316 

368 

421 

473 


496 

50 

99 

149 

198 

24S 

298 

347 

397 

446 


470 


47 

94 

141 

188 

235 

282 

329 

376 

423 


684 


68 

137 

205 

274 

342 

410 

479 

547 

616 


634 


63 

127 

190 

254 

317 

380 

444 

507 

571 


590 


59 

118 

177 

236 

295 

354 

413 

472 

531 


553 


55 

111 

166 

221 

276 

332 

387 

442 

498 


520 


52 

104 

156 

208 

260 

312 

364 

416 

468 


491 


49 

98 

147 

196 

246 

295 

344 

393 

442 


464 


46 

93 

139 

186 

232 

278 

325 

371 

418 


673 

~67~ 

135 

202 

269 

336 

404 

471 

538 

606 


625 

~62 

125 

188 

250 

312 

375 

438 

500 

562 


583 

~58 

117 

175 

233 

292 

350 

408 

466 

525 


546 

55 

109 

164 

218 

273 

328 

382 

437 

491 


514 


51 

103 

154 

206 

257 

308 

360 

411 

463 

485 


48 

97 

146 

194 

242 

291 

340 

388 

436 


460 


89 ° 


[ 36 ] 









































































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 1 


Prop. Parts 


Use the proportional 
parts of the number below 
or at the left which is nearest 
to the actual tabular differ¬ 
ence; the error will be at 
most 2 units in the last place. 
For greater accuracy, use 
Table X on page 83. 



455 

450 

446 

445 

441 

1 

46 

45 

45 

44 

44 

2 

91 

90 

89 

89 

88 

3 

136 

135 

134 

134 

132 

4 

182 

180 

178 

178 

176 

5 

228 

225 

223 

222 

220 

6 

273 

270 

268 

267 

265 

7 

318 

315 

312 

312 

309 

8 

364 

360 

357 

356 

353 

9 

410 

405 

401 

400 

397 


437 

436 

433 

432 

428 

1 

44 

44 

43 

43 

43 

2 

87 

87 

87 

86 

86 

3 

131 

131 

130 

130 

128 

4 

175 

174 

173 

173 

171 

5 

218 

218 

216 

216 

214 

6 

262 

262 

260 

259 

257 

7 

306 

305 

303 

302 

300 

8 

350 

349 

346 

346 

342 

9 

393 

392 

390 

389 

385 


427 

424 

420 

419 

416 

1 

43 

42 

42 

42 

42 

2 

85 

85 

84 

84 

83 

3 

128 

127 

126 

126 

125 

4 

171 

170 

168 

168 

166 

5 

214 

212 

210 

210 

208 

6 

256 

254 

252 

251 

250 

7 

299 

297 

294 

293 

291 

8 

342 

339 

336 

335 

333 

9 

384 

382 

378 

377 

374 


412 

411 

408 

404 

401 

i 

41 

41 

41 

40 

40 

2 

82 

82 

82 

81 

80 

3 

124 

123 

122 

121 

120 

4 

165 

164 

163 

162 

160 

6 

206 

206 

204 

202 

200 

6 

247 

247 

245 

242 

241 

7 

288 

288 

286 

283 

281 

8 

330 

329 

326 

323 

321 

9 

371 

370 

367 

364 

361 


397 

396 

393 

390 

386 

i 

40 

40 

39 

39 

39 

2 

79 

79 

79 

78 

77 

3 

119 

119 

118 

117 

116 

4 

159 

158 

157 

156 

154 

5 

198 

198 

196 

195 

193 

6 

238 

238 

236 

234 

232 

7 

278 

277 

275 

273 

270 

8 

318 

317 

314 

312 

309 

9 

357 

356 

354 

351 

347 


383 

382 

380 

379 

376 

1 

38 

38 

38 

38 

38 

2 

77 

76 

76 

76 

75 

3 

115 

115 

114 

114 

113 

4 

153 

153 

152 

152 

150 

5 

192 

191 

190 

190 

188 

6 

230 

229 

228 

227 

226 

7 

268 

267 

266 

265 

263 

8 

306 

306 

304 

303 

301 

9 

345 

344 

342 

341 

338 


373 

370 

369 

367 

363 

i 

37 

37 

37 

37 

36 

2 

75 

74 

74 

73 

73 

3 

112 

111 

111 

110 

109 

4 

149 

148 

148 

147 

145 

5 

186 

185 

184 

184 

182 

6 

224 

222 

221 

220 

218 

7 

261 

259 

258 

257 

254 

8 

298 

296 

295 

294 

290 

9 

336 

333 

332 

330 

327 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


L Sin * 


8.24 186 


8.24 

8.25 

8.26 
8.26 
8.27 


903 

609 

304 

988 

661 


8.28 324 

8.28 977 

8.29 621 

8.30 255 
8.30 879 


8.31 

8.32 

8.32 

8.33 
8.33 


495 

103 

702 

292 

875 


8.34 450 

8.35 018 

8.35 578 

8.36 131 
8.36 678 


8.37 217 

8.37 750 

8.38 276 

8.38 796 

8.39 310 

8.39 818 

8.40 320 

8.40 816 

8.41 307 
8.41 792 


8.42 272 

8.42 746 

8.43 216 

8.43 680 

8.44 139 

8.44 594 

8.45 044 
8.45 489 

8.45 930 

8.46 366 


8.46 799 

8.47 226 

8.47 650 

8.48 069 
8.48 485 

8.48 896 

8.49 304 

8.49 708 

8.50 108 
8.50 504 


8.50 897 

8.51 287 

8.51 673 

8.52 055 
8.52 434 

8.52 810 

8.53 183 
8.53 552 

8.53 919 

8.54 282 


717 

706 

695 

684 

673 

663 

653 

644 

634 

624 

616 

608 

599 

590 

583 

575 

568 

560 

553 

547 

539 

533 

526 

520 

514 

508 

502 

496 

491 

485 

480 

474 

470 

464 

459 

455 

450 

445 

441 

436 

433 

427 

424 

419 

416 

411 

408 

404 

400 

396 

393 

390 

386 

382 

379 

376 

373 

369 

367 

363 


L Cos * 


L Tan * 


8.24 192 

8.24 910 

8.25 616 

8.26 312 

8.26 996 

8.27 669 

8.28 332 

8.28 986 

8.29 629 

8.30 263 
8.30 888 


8.31 

8.32 

8.32 

8.33 
8.33 


505 

112 

711 

302 

886 


8.34 461 

8.35 029 

8.35 590 

8.36 143 
8.36 689 


8.37 229 

8.37 762 

8.38 289 

8.38 809 

8.39 323 

8.39 832 

8.40 334 

8.40 830 

8.41 321 
8.41 807 


8.42 287 

8.42 762 

8.43 232 

8.43 696 

8.44 156 

8.44 611 

8.45 061 
8.45 507 

8.45 948 

8.46 385 


8.46 817 

8.47 245 

8.47 669 

8.48 089 
8.48 505 

8.48 917 

8.49 325 

8.49 729 

8.50 130 
8.50 527 


8.50 920 

8.51 310 

8.51 696 

8.52 079 
8.52 459 

8.52 835 

8.53 208 
8.53 578 

8.53 945 

8.54 308 


c d 


718 

706 

696 

684 

673 

663 

654 

643 

634 

625 

617 

607 

599 

591 

584 

575 

568 

561 

553 

546 

540 

533 

527 

520 

514 

509 

502 

496 

491 

486 

480 

475 

470 

464 

460 

455 

450 

446 

441 

437 

432 

428 

424 

420 

416 

412 

408 

404 

401 

397 

393 

390 

386 

383 

380 

376 

373 

370 

367 

363 


L Cot * c d 


L Cot 


1.75 808 


1.75 

1.74 

1.73 

1.73 


090 

384 

688 

004 


1.72 331 

1.71 668 
1.71 014 
1.70 371 
1.69 737 
1.69 112 


1.68 495 
1.67 888 
1.67 289 
1.66 698 
1.66 114 

1.65 539 
1.64 971 
1.64 410 
1.63 857 
1.63 311 


1.62 771 
1.62 238 
1.61 711 
1.61 191 
1.60 677 


1.60 

1.59 

1.59 

1.58 

1.58 


168 

666 

170 

679 

193 


1.57 713 
1.57 238 
1.56 768 
1.56 304 
1.55 844 

1.55 389 
1.54 939 
1.54 493 
1.54 052 
1.53 615 


1.53 183 
1.52 755 
1.52 331 
1.51 911 
1.51 495 

1.51 083 
1.50 675 
1.50 271 
1.49 870 
1.49 473 


1.49 080 
1.48 690 
1.48 304 
1.47 921 
1.47 541 

1.47 165 
1.46 792 
1.46 422 
1.46 055 
1.45 692 


L Cos * 


9.99 993 

993 
993 
993 
992 
9.99 992 

992 
992 
992 
991 
9.99 991 


991 
990 
990 
990 
9.99 990 

989 
989 
989 
989 
9.99 988 


988 
988 
987 
987 
9.99 987 

986 
986 
986 
985 
9.99 985 


985 
984 
984 
984 
9.99 983 

983 
983 
982 
982 
9.99 982 


981 
981 
981 
980 
9.99 980 

979 
979 
979 
978 
9.99 978 


L Tan 


977 
977 
977 
976 
9.99 976 

975 
975 
974 
974 
9.99 974 


L Sin * 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


88 ' 


[ 37 ] 





























































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS —2 C 


/ 

L Sin* 

d 

L Tan * 

c d 

L Cot 

L Cos * 


0 

8.54 282 


8.54 308 


1.45 692 

9.99 974 

60 

1 

8.54 642 

360 

8.54 669 

361 

1.45 331 

973 

59 

2 

8.54 999 

357 

8.55 027 

358 

1.44 973 

973 

58 

3 

8.55 354 

355 

382 

355 

Q CO 

618 

972 

57 

4 

8.55 705 

351 

8.55 734 

oo/ 

1.44 266 

972 

56 

5 

8.56 054 

349 

8.56 083 

349 

1.43 917 

9.99 971 

55 

6 

400 

346 

429 

346 

571 

971 

54 

7 

8.56 743 

343 

8.56 773 

344 

Q A 1 

1.43 227 

970 

53 

8 

8.57 084 

341 

8.57 114 

041 

1.42 886 

970 

52 

9 

421 

337 

452 

338 

548 

969 

51 

10 

8.57 757 

336 

8 . 57 , 788 

336 

1.42 212 

9.99 969 

50 

11 

8.58 089 

332 

8.58 121 

333 

1.41 879 

968 

49 

12 

419 

330 

451 

330 

549 

968 

48 

13 

8.58 747 


8.58 779 

328 

QOfl 

1.41 221 

967 

47 

14 

8.59 072 

325 

8.59 105 

oZO 

QOQ 

1.40 895 

967 

46 

15 

8.59 395 


8.59 428 

oZo 

1.40 572 

9.99 967 

45 

16 

8.59 715 

320 

8.59 749 

321 

Q1Q 

1.40 251 

966 

44 

17 

8.60 033 


8.60 068 

oiy 

Q1 A 

1.39 932 

966 

43 

18 

349 


384 

OlO 

616 

965 

42 

19 

662 


8.60 698 

314 

Q1 1 

1.39 302 

964 

41 

20 

8.60 973 


8.61 009 

oil 

1.38 991 

9.99 964 

40 

21 

8.61 282 

309 

319 

310 

Q/Y7 

681 

963 

39 

22 

589 


626 

oU/ 

374 

963 

38 

23 

8.61 894 


8.61 931 

305 

QAQ 

1.38 069 

962 

37 

24 

8.62 196 


8.62 234 

oUo 

OA1 

1.37 766 

962 

36 

25 

8.62 497 


8.62 535 

oUl 

1.37 465 

9.99 961 

35 

26 

8.62 795 

298 

8.62 834 

299 

1.37 166 

961 

34 

27 

8.63 091 


8.63 131 

297 

1.36 869 

960 

33 

28 

385 

294 

OOQ 

426 

295 

ono 

574 

960 

32 

29 

678 

zy o 

onn 

8.63 718 

zyz 

oni 

1.36 282 

959 

31 

30 

8.63 968 

zyu 

8.64 009 

zyi 

1.35 991 

9.99 959 

30 

31 

8.64 256 

288 

OQ7 

298 

289 

702 

958 

29 

32 

543 

Zo7 

OQA 

585 

287 

aoe 

415 

958 

28 

33 

8.64 827 

Ziyi 

oqq 

8.64 870 

Zoo 

QQ A 

1.35 130 

957 

27 

34 

8.65 110 

Zoo 

OQ1 

8.65 154 

Zo4 

OQ1 

1.34 846 

956 

26 

35 

8.65 391 

Zol 

8.65 435 

Zol 

1.34 565 

9.99 956 

25 

36 

670 

279 

O T7 

715 

280 

285 

955 

24 

37 

8.65 947 

Z// 

8.65 993 

278 

1.34 007 

955 

23 

38 

8.66 223 

Z/O 

8.66 269 

276 

1.33 731 

954 

22 

39 

497 

274 

979 

543 

274 

457 

954 

21 

40 

8.66 769 

Z/Z 

8.66 816 

Z/O 

1.33 184 

9.99 953 

20 

41 

8.67 039 

270 

9AO 

8.67 087 

271 

1.32 913 

952 

19 

42 

308 

zoy 

Oft7 

356 

269 

644 

952 

18 

43 

575 

Zb7 

624 

268 

376 

951 

17 

44 

8.67 841 

266 

9AQ 

8.67 890 

266 

1.32 110 

951 

16 

45 

8.68 104 

Zoo 

8.68 154 

264 

1.31 846 

9.99 950 

15 

46 

367 

263 

9ftA 

417 

263 

OA1 

583 

949 

14 

47 

627 

ZoU 

OKO 

678 

Zol 

()/*A 

322 

949 

13 

48 

8.68 886 

zoy 

OKQ 

8.68 938 

ZoU 

OCQ 

1.31 062 

948 

12 

49 

8.69 144 

ZOO 

8.69 196 

Zoo 

1.30 804 

948 

11 

50 

8.69 400 

ZOO 

8.69 453 

ZO/ 

1.30 547 

9.99 947 

10 

51 

654 

254 

9 KQ 

708 

255 

292 

946 

9 

52 

8.69 907 

ZOo 

9KO 

8.69 962 

254 

O KO 

1.30 038 

946 

8 

53 

8.70 159 

zoz 

9 fin 

8.70 214 

Z5Z 

OKI 

1.29 786 

945 

7 

54 

409 

ZOU 

9/IQ 

465 

Z51 

535 

944 

6 

55 

8.70 658 

Z4=y 

8.70 714 

249 

1.29 286 

9.99 944 

5 

56 

8.70 905 

247 

8.70 962 

248 

1.29 038 

943 

4 

57 

8.71 151 

246 

C)AA 

8.71 208 

246 

O a e 

1.28 792 

942 

3 

58 

395 

Z44 

a a o 

453 

Z4o 

547 

942 

2 

59 

638 

Z4o 

697 

244 

303 

941 

1 

60 

8.71 880 

242 

8.71 940 

243 

1.28 060 

9.99 940 

0 


L Cos* 

d 

L Cot* 

C d 

L Tan 

L Sin* 

f 


o 



Prop. 

Parts 

Subtract 

10 from each 

entry in the columns marked 
with 

See opposite page for ad- 

ditional tables. 



361 

360 

358 

1 

36 

36 

36 

2 

72 

72 

72 

3 

108 

108 

107 

4 

144 

144 

143 

5 

180 

180 

179 

6 

217 

216 

215 

7 

253 

252 

251 

8 

289 

288 

286 

9 

325 

324 

322 


357 

3fc5 

352 

1 

36 

36 

35 

2 

71 

71 

70 

3 

107 

106 

106 

4 

143 

142 

141 

5 

178 

178 

176 

6 

214 

213 

211 

7 

250 

248 

246 

8 

286 

284 

282 

9 

321 

320 

317 


351 

349 

346 

1 

35 

35 

35 

2 

70 

70 

69 

3 

105 

105 

104 

4 

140 

140 

138 

5 

176 

174 

173 

6 

211 

209 

208 

7 

246 

244 

242 

8 

281 

279 

277 

9 

316 

314 

311 


344 

343 

341 

1 

34 

34 

34 

2 

69 

69 

68 

3 

103 

103 

102 

4 

138 

137 

136 

5 

172 

172 

170 

6 

206 

206 

205 

7 

241 

240 

239 

8 

275 

274 

273 

9 

310 

309 

307 


338 

337 

336 

1 

34 

34 

34 

2 

68 

67 

67 

3 

101 

101 

101 

4 

135 

135 

134 

5 

169 

168 

168 

6 

203 

202 

202 

7 

237 

236 

235 

8 

270 

270 

269 

9 

304 

303 

302 


333 

332 

330 

1 

33 

33 

33 

2 

67 

66 

66 

3 

100 

100 

99 

4 

133 

133 

132 

5 

166 

166 

165 

6 

200 

199 

198 

7 

233 

232 

231 

8 

266 

266 

264 

9 

300 

299 

297 


87 c 


[ 38 ] 





























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS 
Proportional Parts for 2° and 87° 



328 

326 

325 

323 

321 

320 


319 

318 

316 

314 

313 

311 


1 

33 

33 

32 

32 

32 

32 

1 

32 

32 

32 

31 

31 

31 

1 

2 

66 

65 

65 

65 

64 

64 

2 

64 

64 

63 

63 

63 

62 

2 

3 

98 

98 

98 

97 

96 

96 

3 

96 

95 

95 

94 

94 

93 

3 

4 

131 

130 

130 

129 

128 

128 

4 

128 

127 

126 

126 

125 

124 

4 

5 

164 

163 

162 

162 

160 

160 

5 

160 

159 

158 

157 

156 

156 

5 

6 

197 

196 

195 

194 

193 

192 

6 

191 

191 

190 

188 

188 

187 

6 

7 

230 

228 

228 

226 

225 

224 

7 

223 

223 

221 

220 

219 

218 

7 

8 

262 

261 

260 

258 

257 

256 

8 

255 

254 

253 

251 

250 

249 

8 

9 

295 

293 

292 

291 

289 

288 

9 

287 

286 

284 

283 

282 

280 

9 


310 

309 

307 

305 

303 

302 


301 

299 

298 

297 

296 

295 


1 

31 

31 

31 

30 

30 

30 

1 

30 

30 

30 

30 

30 

30 

1 

2 

62 

62 

61 

61 

61 

60 

2 

60 

60 

60 

59 

59 

59 

2 

3 

93 

93 

92 

92 

91 

91 

3 

90 

90 

89 

89 

89 

88 

3 

4 

124 

124 

123 

122 

121 

121 

4 

120 

120 

119 

119 

118 

118 

4 

5 

155 

154 

154 

152 

152 

151 

5 

150 

150 

149 

148 

148 

148 

5 

6 

186 

185 

184 

183 

182 

181 

6 

181 

179 

179 

178 

178 

177 

6 

7 

217 

216 

215 

214 

212 

211 

7 

211 

209 

209 

208 

207 

206 

7 

8 

248 

247 

246 

244 

242 

242 

8 

241 

239 

238 

238 

237 

236 

8 

9 

279 

278 

276 

274 

273 

272 

9 

271 

269 

268 

267 

266 

266 

9 


294 

293 

292 

291 

290 

289 


288 

287 

285 

284 

283 

281 


1 

29 

29 

29 

29 

29 

29 

1 

29 

29 

28 

28 

28 

28 

1 

2 

59 

59 

58 

58 

58 

58 

2 

58 

57 

57 

57 

57 

56 

2 

3 

88 

88 

88 

87 

87 

87 

3 

86 

86 

86 

85 

85 

84 

3 

4 

118 

117 

117 

116 

116 

116 

4 

115 

115 

114 

114 

113 

112 

4 

5 

147 

146 

146 

146 

145 

144 

5 

144 

144 

142 

142 

142 

140 

5 

6 

176 

176 

175 

175 

174 

173 

6 

173 

172 

171 

170 

170 

169 

6 

7 

206 

205 

204 

204 

203 

202 

7 

202 

201 

200 

199 

198 

197 

7 

8 

235 

234 

234 

233 

232 

231 

8 

230 

230 

228 

227 

226 

225 

8 

9 

265 

264 

263 

262 

261 

260 

9 

259 

258 

256 

256 

255 

253 

9 


280 

279 

278 

277 

276 

274 


273 

272 

271 

270 

269 

263 


1 

28 | 

28 

28 

28 

28 

27 

1 

27 

27 

27 

27 

27 

27 

X 

2 

56 

56 

56 

55 

55 

55 

2 

55 

54 

54 

54 

54 

54 

2 

3 

84 

84 

83 

83 

83 

82 

3 

82 

82 

81 

81 j 

81 

80 

3 

4 

112 

112 

111 

111 

110 

110 

4 

109 

109 

108 

108 

108 

107 

4 

5 

140 

140 

139 

138 

138 

137 

5 

136 

136 

136 

135 

134 

134 

5 

6 

168 

167 

167 

166 

166 

164 

6 

164 

163 

163 

162 

161 

161 

6 

7 

196 

195 

195 

194 

193 

192 

7 

191 

190 

190 

189 

188 

188 

7 

8 

224 

223 

222 

222 

221 

219 

8 

218 

218 

217 

216 

215 

214 

8 

9 

252 

251 

250 

249 

248 

247 

9 

246 

245 

244 

243 

242 

241 

9 


267 

266 

264 

263 

261 

260 


259 

258 

257 

256 

255 

254 


1 

27 

27 

26 

26 

26 

26 

1 

26 

26 

26 

26 

26 

25 

1 

2 

53 

53 

53 

53 

52 

52 

2 

52 

52 

51 

51 

51 

51 

2 

3 

80 

80 

79 

79 

78 

78 

3 

78 

77 

77 

77 

76 

76 

3 

4 

107 

106 

106 

105 

104 

104 

4 

104 

103 

103 

102 

102 

102 

4 

5 

134 

133 

132 

132 

130 

130 

6 

130 

129 

128 

128 

128 

127 

5 

6 

160 

160 

158 

158 

157 

156 

6 

155 

155 

154 

154 

153 

152 

6 

7 

187 

186 

185 

184 

183 

182 

7 

181 

181 

180 

179 

178 

178 

7 

8 

214 

213 

211 

210 

209 

208 

8 

207 

206 

206 

205 

204 

203 

8 

9 

240 

239 

238 

237 

235 

234 

9 

233 

232 

231 

230 

230 

229 

9 


253 

252 

251 

250 

249 

248 


247 

246 

245 

244 

243 

242 


1 

25 

25 

25 

25 

25 

25 

1 

25 

25 

24 

24 

24 

24 

1 

2 

51 

50 

50 

50 

50 

50 

2 

49 

49 

49 

49 

49 

48 

2 

3 

76 

76 

75 

75 

75 

74 

3 

74 

74 

74 

73 

73 

73 

3 

4 

101 

101 

100 

100 

100 

99 

4 

99 

98 

98 

98 

97 

97 

4 

5 

126 

126 

126 

125 

124 

124 

5 

124 

123 

122 

122 

122 

121 

5 

6 

152 

151 

151 

150 

149 

149 

6 

148 

148 

147 

146 

146 

145 

6 

7 

177 

176 

176 

175 

174 

174 

7 

173 

172 

172 

171 

170 

169 

7 

8 

202 

202 

201 

200 

199 

198 

8 

198 

197 

196 

195 

194 

194 

8 

9 

228 

227 

226 

225 

224 

223 

9 

222 

221 

220 

220 

219 

218 

9 


[ 39 ] 

































































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 3' 


/ 

L Sin * 

d 

L Tan * 

c d 

L Cot 

L Cos * 


0 

8.71 880 


8.71 940 


1.28 060 

9.99 940 

60 

1 

8.72 120 

240 

8.72 181 

241 

1.27 819 

940 

59 

2 

359 

239 

420 

239 

580 

939 

58 

3 

597 

238 

659 

239 

341 

938 

57 

4 

8.72 834 

237 

8.72 896 

237 

1.27 104 

938 

56 

5 

8.73 069 

23 ^ 

8.73 132 

236 

1.26 868 

9.99 937 

55 

6 

303 

234 

366 

234 

634 

936 

54 

7 

535 

232 

600 

234 

400 

936 

53 

8 

767 

232 

8.73 832 

232 

1.26 168 

935 

52 

9 

8.73 997 

230 

8.74 063 

231 

1.25 937 

934 

51 

10 

8.74 226 

229 

8.74 292 

229 

1.25 708 

9.99 934 

50 

11 

454 

228 

521 

229 

479 

933 

49 

12 

680 

226 

748 

227 

252 

932 

48 

13 

8.74 906 

226 

8.74 974 

226 

1.25 026 

932 

47 

14 

8.75 130 

224 

8.75 199 

22 o 

1.24 801 

931 

46 

15 

8.75 353 

223 

8.75 423 

224 

1.24 577 

9.99 930 

45 

16 

575 

222 

645 

222 

355 

929 

44 

17 

8.75 795 

220 

8.75 867 

222 

1.24 133 

929 

43 

18 

8.76 015 

220 

8.76 087 

220 

1.23 913 

928 

42 

19 

234 

219 

306 

219 

694 

927 

41 

20 

8.76 451 

217 

8.76 525 

219 

1.23 475 

9.99 926 

40 

21 

667 

216 

742 

217 

258 

926 

39 

22 

8.76 883 

216 

8.76 958 

216 

1.23 042 

925 

38 

23 

8.77 097 

214 

8.77 173 

2 X 5 

1.22 827 

924 

37 

24 

310 

213 

387 

214 

613 

923 

36 

25 

8.77 522 

212 

8.77 600 

213 

1.22 400 

9.99 923 

35 

26 

733 

211 

8.77 811 

211 

1.22 189 

922 

34 

27 

8.77 943 

210 

8.78 022 

211 

1.21 978 

921 

33 

28 

8.78 152 

209 

232 

210 

768 

920 

32 

29 

360 

208 

441 

209 

559 

920 

31 

30 

8.78 568 

208 

8.78 649 

208 

1.21 351 

9.99 919 

30 

31 

774 

206 

8.78 855 

206 

1.21 145 

918 

29 

32 

8.78 979 

205 

8.79 061 

206 

1.20 939 ' 

917 

28 

33 

8.79 183 

204 

266 

20 o 

734 

917 

27 

34 

386 

203 

470 

204 

530 

916 

26 

35 

8.79 588 

202 

8.79 673 

203 

1.20 327 

9.99 915 

25 

36 

789 

201 

8.79 875 

202 

1.20 125 

914 

24 

37 

8.79 990 

201 

8.80 076 

201 

1.19 924 

913 

23 

38 

8.80 189 

199 

277 

201 

723 

913 

22 

39 

388 

199 

476 

199 

524 

912 

21 

40 

8.80 585 

197 

8.80 674 

198 

1.19 326 

9.99 911 

20 

41 

782 

197 

8.80 872 

198 

1.19 128 

910 

19 

42 

8.80 978 

196 

8.81 068 

196 

1.18 932 

909 

18 

43 

8.81 173 

195 

264 

196 

736 

909 

17 

44 

367 

194 

459 

195 

541 

908 

16 

45 

8.81 560 

193 

8.81 653 

194 

1.18 347 

9.99 907 

15 

46 

752 

192 

8.81 846 

193 

1.18 154 

906 

14 

47 

8.81 944 

192 

8.82 038 

192 

1.17 962 

905 

13 

48 

8.82 134 

190 

230 

192 

770 

904 

12 

49 

324 

190 

420 

190 

5 S 0 

904 

11 

50 

8.82 513 

189 

8.82 610 

190 

1.17 390 

9.99 903 

10 

51 

701 

188 

799 

189 

201 

902 

9 

52 

8.82 888 

187 

8.82 987 

188 

1.17 013 

901 

8 

53 

8.83 075 

187 

8.83 175 

188 

1.16 825 

900 

7 

54 

261 


361 


639 

899 

6 

55 

8.83 446 

185 

8.83 547 

186 

1.16 453 

9.99 898 

5 

56 

630 

184 

732 

185 

268 

898 

4 

57 

813 


8.83 916 


1.16 084 

897 

3 

58 

8.83 996 


8.84 100 


1.15 900 

896 

2 

59 

8.84 177 


282 

189 

718 

895 

1 

60 

8.84 358 


8.84 464 


1.15 536 

9.99 894 

0 


L Cos * 

d 

L Cot * 

| c d 

L Tan 

L Sin * 

/ 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with 

See opposite page for ad¬ 
ditional tables. 



241 

240 

1 

24.1 

24 

2 

48.2 

48 

3 

72.3 

72 

4 

96.4 

96 

5 

120.5 

120 

6 

144.6 

144 

7 

168.7 

168 

8 

192.8 

192 

9 

216.9 

216 



239 

238 

1 

23.9 

23.8 

2 

47.8 

47.6 

3 

71.7 

71.4 

4 

95.6 

95.2 

5 

119.5 

119.0 

6 

143.4 

142.8 

7 

167.3 

166.6 

8 

191.2 

190.4 

9 

215.1 

214.2 



237 

1 

23.7 

2 

47.4 

3 

71.1 

4 

94.8 

5 

118.5 

6 

142.2 

7 

165.9 

8 

189.6 

9 

213.3 



236 

235 

1 

23.6 

23.5 

2 

47.2 

47.0 

3 

70.8 

70.5 

4 

94.4 

94.0 

5 

118.0 

117.5 

6 

141.6 

141.0 

7 

165.2 

164.5 

8 

188.8 

188.0 

9 

212.4 

211.5 


[ 40 ] 


86 ' 
















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS 
Proportional Parts for 3° and 86° 



234 

232 

231 

230 


229 

228 

227 

226 


1 

23.4 

23.2 

23.1 

23 

1 

22.9 

22.8 

22.7 

22.6 

1 

2 

46.8 

46.4 

46.2 

46 

2 

45.8 

45.6 

45.4 

45.2 

2 

3 

70.2 

69.6 

69.3 

69 

3 

68.7 

68.4 

68.1 

67.8 

3 

4 

93.6 

92.8 

92.4 

92 

4 

91.6 

91.2 

90.8 

90.4 

4 

5 

117.0 

116.0 

115.5 

115 

5 

114.5 

114.0 

113.5 

113.0 

5 

6 

140.4 

139.2 

138.6 

138 

6 

137.4 

136.8 

136.2 

135.6 

6 

7 

163.8 

162.4 

161.7 

161 

7 

160.3 

159.6 

158.9 

158.2 

7 

8 

187.2 

185.6 

184.8 

184 

8 

183.2 

182.4 

181.6 

180.8 

8 

9 

210.6 

208.8 

207.9 

207 

9 

206.1 

205.2 

204.3 

203.4 

9 


225 

224 

223 

222 


220 

219 

217 

216 


1 

22.5 

22.4 

22.3 

22.2 

1 

22 

21.9 

21.7 

21.6 

1 

2 

45.0 

44.8 

44.6 

44.4 

2 

44 

43.8 

43.4 

43.2 

2 

3 

67.5 

67.2 

66.9 

66.6 

3 

66 

65.7 

65.1 

64.8 

3 

4 

90.0 

89.6 

89.2 

88.8 

4 

88 

87.6 

86.8 

86.4 

4 

5 

112.5 

112.0 

111.5 

111.0 

5 

110 

109.5 

108.5 

108.0 

5 

6 

135.0 

134.4 

133.8 

133.2 

6 

132 

131.4 

130.2 

129.6 

6 

7 

157.5 

156.8 

156.1 

155.4 

7 

154 

153.3 

151.9 

151.2 

7 

8 

180.0 

179.2 

178.4 

177.6 

8 

176 

175.2 

173.6 

172.8 

8 

9 

202.5 

201.6 

200.7 

199.8 

9 

198 

197.1 

195.3 

194.4 

9 


215 

214 

213 

212 


211 

210 

209 

208 


1 

21.5 

21.4 

21.3 

21.2 

1 

21.1 

21 

20.9 

20.8 

1 

2 

43.0 

42.8 

42.6 

42.4 

2 

42.2 

42 

41.8 

41.6 

2 

3 

64.5 

64.2 

63.9 

63.6 

3 

63.3 

63 

62.7 

62.4 

3 

4 

86.0 

85.6 

85.2 

84.8 

4 

84.4 

84 

83.6 

83.2 

4 

5 

107.5 

107.0 

106.5 

106.0 

5 

105.5 

105 

104.5 

104.0 

5 

6 

129.0 

128.4 

127.8 

127.2 

6 

126.6 

126 

125.4 

124.8 

6 

7 

150.5 

149.8 

149.1 

148.4 

7 

147.7 

147 

146.3 

145.6 

7 

8 

172.0 

171.2 

170.4 

169.6 

8 

168.8 

168 

167.2 

166.4 

8 

9 

193.5 

192.6 

191.7 

190.8 

9 

189.9 

189 

188.1 

187.2 

9 


206 

205 

204 

203 


202 

201 

199 

198 


1 

20.6 

20.5 

20.4 

20.3 

1 

20.2 

20.1 

19.9 

19.8 

1 

2 

41.2 

41.0 

40.8 

40.6 

2 

40.4 

40.2 

39.8 

39.6 

2 

3 

61.8 

61.5 

61.2 

60.9 

3 

60.6 

60.3 

59.7 

59.4 

3 

4 

82.4 

82.0 

81.6 

81.2 

4 

80.8 

80.4 

79.6 

79.2 

4 

5 

103.0 

102.5 

102.0 

101.5 

5 

101.0 

100.5 

99.5 

99.0 

5 

6 

123.6 

123.0 

122.4 

121.8 

6 

121.2 

120.6 

119.4 

118.8 

6 

7 

144.2 

143.5 

142.8 

142.1 

7 

141.4 

140.7 

139.3 

138.6 

7 

8 

164.8 

164.0 

163.2 

162.4 

8 

161.6 

160.8 

159.2 

158.4 

8 

9 

185.4 

184.5 

183.6 

182.7 

9 

181.8 

180.9 

179.1 

178.2 

9 


197 

196 

195 

194 


193 

192 

190 

189 


1 

19.7 

19.6 

19.5 

19.4 

1 

19.3 

19.2 

19 

18.9 

1 

2 

39.4 

39.2 

39.0 

38.8 

2 

38.6 

38.4 

38 

37.8 

2 

3 

59.1 

58.8 

58.5 

58.2 

3 

57.9 

57.6 

57 

56.7 

3 

4 

78.8 

78.4 

78.0 

77.6 

4 

77.2 

76.8 

76 

75.6 

4 

5 

98.5 

98.0 

97.5 

97.0 

5 

96.5 

96.0 

95 

94.5 

5 

6 

118.2 

117.6 

117.0 

116.4 

6 

115.8 

115.2 

114 

113.4 

6 

7 

137.9 

137.2 

136.5 

135.8 

7 

135.1 

134.4 

133 

132.3 

7 

8 

157.6 

156.8 

156.0 

155.2 

8 

154.4 

153.6 

152 

151.2 

8 

9 

177.3 

176.4 

175.5 

174.6 

9 

173.7 

172.8 

171 

170.1 

9 


188 

187 

186 

185 


184 

183 

182 

181 


1 

18.8 

18.7 

18.6 

18.5 

1 

18.4 

18.3 

18.2 

18.1 

1 

2 

37.6 

37.4 

37.2 

37.0 

2 

36.8 

36.6 

36.4 

36.2 

2 

3 

56.4 

56.1 

55.8 

55.5 

3 

55.2 

54.9 

54.6 

54.3 

3 

4 

75.2 

74.8 

74.4 

74.0 

4 

73.6 

73.2 

72.8 

72.4 

4 

5 

94.0 

93.5 

93.0 

92.5 

5 

92.0 

91.5 

91.0 

90.5 

5 

6 

112.8 

112.2 

111.6 

111.0 

6 

110.4 

109.8 

109.2 

108.6 

6 

7 

131.6 

130.9 

130.2 

129.5 

7 

128.8 

128.1 

127.4 

126.7 

7 

8 

150.4 

149.6 

148.8 

148.0 

8 

147.2 

146.4 

145.6 

144.8 

8 

9 

169.2 

168.3 

167.4 

166.5 

9 

165.6 

164.7 

163.8 

162.9 

9 


[ 41 ] 

















































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 4‘ 



[ 42 ] 


85 







































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 5° 



84 ° 


[ 43 ] 
















































































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 6' 


L Sin * 


9.01 923 

9.02 043 
163 
283 
402 
9.02 520 

639 
757 
874 
9.02 992 
9.03 109 


226 
342 
458 
574 
9.03 690 

805 
9.03 920 
9.04 034 
149 
9.04 262 


376 
490 
603 
715 
9.04 828 

9.04 940 
9.05 052 
164 
275 
9.05 386 


497 
607 
717 
827 
9.05 937 

9.06 046 
155 
264 
372 
9.06 481 


589 
696 
804 
9.06 911 
9.07 018 

124 
231 
337 
442 
9.07 548 


653 
758 
863 
9.07 968 
9.08 072 

176 
280 
383 
486 
9.08 589 


L Cos * 


120 

120 

120 

119 

118 

119 

118 

117 

118 
117 

117 

116 

116 

116 

116 

115 

115 

114 

115 

113 

114 
114 
113 
112 
113 

112 

112 

112 

111 

111 

111 

110 

110 

110 

110 

109 

109 

109 

108 

109 

108 

107 

108 
107 
107 

106 

107 

106 

105 

106 

105 

105 

105 

105 

104 

104 

104 

103 

103 

103 


L Tan * 


9.02 162 

283 
404 
525 
645 
9.02 766 

9.02 885 
9.03 005 
124 
242 
9.03 361 


479 
597 
714 
832 
9.03 948 

9.04 065 
181 
297 
413 
9.04 528 


643 
758 
873 
9.04 987 
9.05 101 

214 
328 
441 
553 
9.05 666 


778 
9.05 890 
9.06 002 
113 
9.06 224 

335 
445 
556 
666 
9.06 775 


885 
9.06 994 
9.07 103 
211 
9.07 320 

428 
536 
643 
751 
9.07 858 


9.07 964 
9.08 071 
177 
283 
9.08 389 

495 
600 
705 
810 
9.08 914 


c d 


121 

121 

121 

120 

121 

119 

120 
119 
118 
119 

118 

118 

117 

118 
116 

117 

116 

116 

116 

115 

115 

115 

115 

114 

114 

113 

114 
113 
112 
113 

112 

112 

112 

111 

111 

111 

110 

111 

110 

109 

110 
109 
109 
108 
109 

108 

108 

107 

108 
107 

106 

107 

106 

106 

106 

106 

105 

105 

105 

104 


L Cot 

L Cos * 


0.97 838 

9.99 761 

60 

717 

760 

59 

596 

759 

58 

475 

757 

57 

355 

756 

56 

0.97 234 

9.99 755 

55 

0.97 115 

753 

54 

0.96 995 

752 

53 

876 

751 

52 

758 

749 

51 

0.96 639 

9.99 748 

50 

521 

747 

49 

403 

745 

48 

286 

744 

47 

168 

742 

46 

0.96 052 

9.99 741 

45 

0.95 935 

740 

44 

819 

738 

43 

703 

737 

42 

587 

736 

41 

0.95 472 

9.99 734 

40 

357 

733 

39 

242 

731 

38 

127 

730 

37 

0.95 013 

728 

36 

0.94 899 

9.99 727 

35 

786 

726 

34 

672 

724 

33 

559 

723 

32 

447 

721 

31 

0.94 334 

9.99 720 

30 

222 

718 

29 

0.94 110 

717 

28 

0.93 998 

716 

27 

887 

714* 

26 

0.93 776 

9.99 713 

25 

665 

711 

24 

555 

710 

23 

444 

708 

22 

334 

707 

21 

0.93 225 

9.99 705 

20 

115 

704 

19 

0.93 006 

702 

18 

0.92 897 

701 

17 

789 

699 

16 

0.92 680 

9.99 698 

15 

572 

696 

14 

464 

695 

13 

357 

693 

12 

249 

692 

11 

0.92 142 

9.99 690 

10 

0.92 036 

689 

9 

0.91 929 

687 

8 

823 

686 

7 

717 

684 

6 

0.91 611 

9.99 683 

5 

505 

681 

4 

400 

680 

3 

295 

678 

2 

190 

677 

1 

0.91 086 

9.99 675 

0 

L Tan 

L Sin * 

/ 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with 



121 

120 

119 

1 

12.1 

12 

11.9 

2 

24.2 

24 

23.8 

3 

36.3 

36 

35.7 

4 

48.4 

48 

47.6 

5 

60.5 

60 

59.5 

6 

72.6 

72 

71.4 

7 

84.7 

84 

83.3 

8 

96.8 

96 

95.2 

9 

108.9 

108 

107.1 



118 

117 

116 

1 

11.8 

11.7 

11.6 

2 

23.6 

23.4 

23.2 

3 

35.4 

35.1 

34.8 

4 

47.2 

46.8 

46.4 

5 

59.0 

58.5 

58.0 

6 

70.8 

70.2 

69.6 

7 

82.6 

81.9 

81.2 

8 

94.4 

93.6 

92.8 

9 

106.2 

105.3 

104.4 



115 

114 

113 

1 

11.5 

11.4 

11.3 

2 

23.0 

22.8 

22.6 

3 

34.5 

34.2 

33.9 

4 

46.0 

45.6 

45.2 

5 

57.5 

57.0 

56.5 

6 

69.0 

68.4 

67.8 

7 

80.5 

79.8 

79.1 

8 

92.0 

91.2 

90.4 

9 

103.5 

102.6 

101.7 



112 

111 

110 

1 

11.2 

11.1 

11 

2 

22.4 

22.2 

22 

3 

33.6 

33.3 

33 

4 

44.8 

44.4 

44 

5 

56.0 

55.5 

55 

6 

67.2 

66.6 

66 

7 

78.4 

77.7 

77 

8 

89.6 

88.8 

88 

9 

100.8 

99.9 

99 



109 

108 

107 

106 

1 

10.9 

10.8 

10.7 

10.6 

2 

21.8 

21.6 

21.4 

21.2 

3 

32.7 

32.4 

32.1 

31.8 

4 

43.6 

43.2 

42.8 

42.4 

5 

54.5 

54.0 

53.5 

53.0 

6 

65.4 

64.8 

64.2 

63.6 

7 

76.3 

75.6 

74.9 

74.2 

8 

87.2 

86.4 

85.6 

84.8 

9 

98.1 

97.2 

96.3 

95.4 


[ 44 ] 


83 ' 






















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 7' 



82 ' 


[ 45 ] 





















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 8' 


/ 

L Sin * 

0 

9.14 356 

1 

445 

2 

535 

3 

624 

4 

714 

5 

9.14 803 

6 

891 

7 

9.14 980 

8 

9.15 069 

9 

157 

10 

9.15 245 

11 

333 

12 

421 

13 

508 

14 

596 

15 

9.15 683 

16 

770 

17 

857 

18 

9.15 944 

19 

9.16 030 

20 

9.16 116 

21 

203 

22 

289 

23 

374 

24 

460 

25 

9.16 545 

26 

631 

27 

716 

28 

801 

29 

886 

30 

9.16 970 

31 

9.17 055 

32 

139 

33 

223 

34 

307 

35 

9.17 391 

36 

474 

37 

558 

38 

641 

39 

724 

40 

9.17 807 

41 

890 

42 

9.17 973 

43 

9.18 055 

44 

137 

45 

9.18 220 

46 

302 

47 

383 

48 

465 

49 

547 

50 

9.18 628 

51 

709 

52 

790 

53 

871 

54 

9.18 952 

55 

9.19 033 

56 

113 

57 

193 

58 

273 

59 

353 

60 

9.19 433 


L Cos * 


L Tan * c d 


9.14 780 

872 

9.14 963 

9.15 054 
145 

9.1£ 236 

327 
417 
508 
598 
9.15 688 


777 

867 

9.15 956 

9.16 046 
9.16 135 

224 
312 
401 
489 
9.16 577 


665 

753 

841 

9.16 928 

9.17 016 

103 
190 
277 
363 
9.17 450 


536 
622 
708 
794 
9.17 880 

9.17 965 

9.18 051 
136 

991 

9.18 306 


391 
475 
560 
644 
9.18 728 

812 

896 

9.18 979 

9.19 063 
9.19 146 


229 
312 
395 
478 
9.19 561 

643 
725 
807 
889 
9.19 971 


L Cot * c d 


L Cot 

L Cos * 


0.85 220 

9.99 575 

60 

128 

574 

59 

0.85 037 

572 

58 

0.84 946 

570 

57 

855 

568 

56 

0.84 764 

9.99 566 

55 

673 

565 

54 

583 

563 

53 

492 

561 

52 

402 

559 

51 

0.84 312 

9.99 557 

50 

223 

556 

49' 

133 

554 

48 

0.84 044 

552 

47 

0.83 954 

550 

46 

0.83 865 

9.99 548 

45 

776 

546 

44 

688 

545 

43 

599 

543 

42 

511 

541 

41 

0.83 423 

9.99 539 

40 

335 

537 

39 

247 

535 

38 

159 

533 

37 

0.83 072 

532 

36 

0.82 984 

9.99 530 

35 

897 

528 

34 

810 

526 

33 

723 

524 

32 

637 

522 

31 

0.82 550 

9.99 520 

30 

464 

518 

29 

378 

517 

28 

292 

515 

27 

206 

513 

26 

0.82 120 

9.99 511 

25 

0.82 035 

509 

24 

0.81 949 

507 

23 

864 

505 

22 

779 

503 

21 

0.81 694 

9.99 501 

20 

609 

499 

19 

525 

497 

18 

440 

495 

17 

356 

494 

16 

0.81 272 

9.99 492 

15 

188 

490 

14 

104 

488 

13 

0.81 021 

486 

12 

0.80 937 

484 

11 

0.80 854 

9.99 482 

10 

771 

480 

9 

688 

478 

8 

605 

476 

7 

522 

474 

6 

0.80 439 

9.99 472 

5 

357 

470 

4 

275 

468 

3 

193 

466 

2 

111 

464 

1 

0.80 029 

9.99 462 

0 

L Tan 

L Sin* 

/ 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with ” 



92 

91 

90 

89 

1 

9.2 

9.1 

9 

8.9 

2 

18.4 

18.2 

18 

17.8 

3 

27.6 

27.3 

27 

26.7 

4 

36.8 

36.4 

36 

35.6 

5 

46.0 

45.5 

45 

44.5 

6 

55.2 

54.6 

54 

53.4 

7 

64.4 

63.7 

63 

62.3 

8 

73.6 

72.8 

72 

71.2 

9 

82.8 

81.9 

81 

80.1 



88 

87 

86 

1 

8.8 

8.7 

8.6 

2 

17.6 

17.4 

17.2 

3 

26.4 

26.1 

25.8 

4 

35.2 

34.8 

34.4 

5 

44.0 

43.5 

43.0 

6 

52.8 

52.2 

51.6 

7 

61.6 

60.9 

60.2 

8 

70.4 

69.6 

68.8 

9 

79.2 

78.3 

77.4 



85 

84 

83 

1 

8.5 

8.4 

8.3 

2 

17.0 

16.8 

16.6 

3 

25.5 

25.2 

24.9 

4 

34.0 

33.6 

33.2 

5 

42.5 

42.0 

41.5 

6 

51.0 

50.4 

49.8 

7 

59.5 

58.8 

58.1 

8 

68.0 

67.2 

66.4 

9 

76.5 

75.6 

74.7 



82 

81 

80 

1 

8.2 

8.1 

8 

2 

16.4 

16.2 

16 

3 

24.6 

24.3 

24 

4 

32.8 

32.4 

32 

5 

41.0 

40.5 

40 

6 

49.2 

48.6 

48 

7 

57.4 

56.7 

56 

8 

65.6 

64.8 

64 

9 

73.8 

72.9 

72 


[ 46 ] 


81 






















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 9 



Prop 

. Parts 



82 

81 

80 

1 

8.2 

8.1 

8 

2 

16.4 

16.2 

16 

3 

24.6 

24.3 

24 

4 

32.8 

32.4 

32 

5 

41.0 

40.5 

40 

6 

49.2 

48.6 

48 

7 

57.4 

56.7 

56 

8 

65.6 

64.8 

64 

9 

73.8 

72.9 

72 


79 

78 

77 

1 

7.9 

7.8 

7.7 

2 

15.8 

15.6 

15.4 

3 

23.7 

23.4 

23.1 

4 

31.6 

31.2 

30.8 

5 

39.5 

39.0 

38.5 

6 

47.4 

46.8 

46.2 

7 

55.3 

54.6 

53.9 

8 

63.2 

62.4 

61.6 

9 

71.1 

70.2 

69.3 


76 

75 

74 

1 

7.6 

7.5 

7.4 

2 

15.2 

15.0 

14.8 

3 

22.8 

22.5 

22.2 

4 

30.4 

30.0 

29.6 

5 

38.0 

37.5 

37.0 

6 

45.6 

45.0 

44.4 

7 

53.2 

52.5 

51.8 

8 

60.8 

60.0 

59.2 

9 

68.4 

67.5 

66.6 


73 

72 

71 

1 

7.3 

7.2 

7.1 

2 

14.6 

14.4 

14.2 

3 

21.9 

21.6 

21.3 

4 

29.2 

28.8 

28.4 

5 

36.5 

36.0 

35.5 

6 

43.8 

43.2 

42.6 

7 

51.1 

50.4 

49.7 

8 

58.4 

57.6 

56.8 

9 

65.7 

64.8 

63.9 


/ 

L Sin* 

d 

L Tan * 

c d 

L Cot 

L Cos* 

0 

9.19 433 


9.19 971 


0.80 029 

9.99 462 

1 

513 

80 

70 

9.20 053 

82 

0.79 947 

460 

2 

592 

/ y 
an 

134 

81 

Of) 

866 

458 

3 

672 

ou 

70 

216 

oZ 

Q I 

784 

456 

4 

751 

/ y 

70 

297 

ol 

Q 1 

703 

454 

5 

9.19 830 

/ y 

9.20 378 

ol 

0.79 622 

9.99 452 

6 

909 

79 

7Q 

459 

81 

541 

450 

7 

9.19 988 

/ y 

70 

540 

81 

Q 1 

460 

448 

8 

9.20 067 

/ y 

7ft 

621 

ol 

on 

379 

446 

9 

145 

4 o 

7ft 

701 

oU 

Q 1 

299 

444 

10 

9.20 223 

4 O 

9.20 782 

ol 

0.79 218 

9.99 442 

11 

302 

79 

7ft 

862 

80 

on 

138 

440 

12 

380 

4 O 

7ft 

9.20 942 

oU 

on 

0.79 058 

438 

13 

458 

4 O 

77 

9.21 022 

oU 

on 

0.78 978 

436 

14 

535 

4 4 

7ft 

102 

oU 

on 

898 

434 

15 

9.20 613 

4 O 

9.21 182 

oU 

0.78 818 

9.99 432 

16 

691 

78 

77 

261 

79 

on 

739 

429 

17 

768 

4 4 

77 

341 

oU 

659 

427 

18 

845 

4 4 

77 

420 

79 

580 

425 

19 

922 

77 

499 

/ y 

70 

501 

423 

20 

9.20 999 

4 4 

9.21 578 

/y 

0.78 422 

9.99 421 

21 

9.21 076 

77 

77 

657 

79 

343 

419 

22 

153 

4 4 

7A 

736 

7y 

7Q 

264 

417 

23 

229 

4 O 

77 

814 

7o 

7Q 

186 

415 

24 

306 

7ft 

893 

/y 

70 

107 

413 

25 

9.21 382 

4 O 

9.21 971 

4 0 

0.78 029 

9.99 411 

26 

458 

76 

9.22 049 

78 

0.77 951 

409 

27 

534 

f O 

7ft 

127 

78 

873 

407 

28 

610 

4 O 

7*% 

205 

78 

70 

795 

404 

29 

685 

4 0 
7ft 

283 

4 0 

7Q 

717 

402 

30 

9.21 761 

4 O 

9.22 361 

7o 

0.77 639 

9.99 400 

31 

836 

75 

7 ft 

438 

77 

7Q 

562 

398 

32 

912 

4 O 

17 jr 

516 

4 0 

484 

396 

33 

9.21 987 

75 

7 ^ 

593 

77 

407 

394 

34 

9.22 062 

4 5 

7 K 

670 

77 

330 

392 

35 

9.22 137 

7 5 

9.22 747 

77 

0.77 253 

9.99 390 

36 

211 

74 

824 

77 

176 

388 

37 

286 

75 

901 

77 

099 

385 

38 

361 

75 

7/1 

9.22 977 

76 

77 

0.77 023 

383 

39 

435 

/ 4 

7/1 

9.23 054 

4 1 

7ft 

0.76 946 

381 

40 

9.22 509 

74 

9.23 130 

7o 

0.76 870 

9.99 379 

41 

583 

74 

206 

76 

794 

377 

42 

657 

74 

283 

77 

717 

375 

43 

731 

74 

359 

76 

641 

372 

44 

805 

74 

435 

76 

565 

370 

45 

9.22 878 

73 

9.23 510 

75 

0.76 490 

9.99 368 

46 

9.22 952 

74 

586 

76 

414 

366 

47 

9.23 025 

73 

661 

75 

339 

364 

48 

098 

73 

737 

76 

263 

362 

49 

171 

73 

812 

75 

188 

359 

50 

9.23 244 

73 

9.23 887 

75 

0.76 113 

9.99 357 

51 

317 

73 

9.23 962 

75 

0.76 038 

355 

52 

390 

73 

9.24 037 

75 

0.75 963 

353 

53 

462 

72 

112 

75 

888 

351 

54 

535 

73 

186 

74 

814 

348 

55 

9.23 607 

72 

9.24 261 

75 

0.75 739 

9.99 346 

56 

679 

72 

335 

74 

665 

344 

57 

752 

73 

410 

75 

590 

342 

58 

823 

71 

484 

74 

516 

340 

59 

895 

72 

558 

74 

442 

337 

60 

9.23 967 

72 

9.24 632 

74 

0.75 368 

9.99 335 


L Cos* 

d 

L Cot* 

c d 

L Tan 

L Sin* 


80 ‘ 


[ 47 ] 























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 10' 



[ 48 ] 


79 ' 














































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 11 



78 ' 


[ 49 ] 

















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 12' 



[ 50 ] 


77 ' 














































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 13 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with “*.” 



58 

57 

56 

1 

5.8 

5.7 

5.6 

2 

11.6 

11.4 

11.2 

3 

17.4 

17.1 

16.8 

4 

23.2 

22.8 

22.4 

5 

29.0 

28.5 

28.0 

6 

34.8 

34.2 

33.6 

7 

40.6 

39.9 

39.2 

8 

46.4 

45.6 

44.8 

9 

52.2 

51.3 

50.4 



55 

54 

53 

1 

5.5 

5.4 

5.3 

2 

11.0 

10.8 

10.6 

3 

16.5 

16.2 

15.9 

4 

22.0 

21.6 

21.2 

5 

27.5 

27.0 

26.5 

6 

33.0 

32.4 

31.8 

7 

38.5 

37.8 

37.1 

8 

44.0 

43.2 

42.4 

9 

49.5 

48.6 

47.7 



52 

51 

1 

5.2 

5.1 

2 

10.4 

10.2 

3 

15.6 

15.3 

4 

20.8 

20.4 

5 

26.0 

25.5 

6 

31.2 

30.6 

7 

36.4 

35.7 

8 

41.6 

40.8 

9 

46.8 

45.9 



4 

3 

2 

1 

0.4 

0.3 

0.2 

2 

0.8 

0.6 

0.4 

3 

.1.2 

0.9 

0.6 

4 

1.6 

1.2 

0.8 

5 

2.0 

1.5 

1.0 

6 

2.4 

1.8 

1.2 

7 

2.8 

2.1 

1.4 

8 

3.2 

2.4 

1.6 

9 

3.6 

2.7 

1.8 



L Sin* 

d 

L Tan* 

c d 

L Cot 

L Cos* 

d 


0 

9.35 209 


9.36 336 


0.63 664 

9.98 872 


60 

1 

263 

54 

55 

394 

58 

KQ 

606 

869 

3 

59 

2 

318 

** 

452 

DO 

548 

867 

2 

58 

3 

373 

oo 

509 

0 / 

P17 

491 

864 

3 

57 

4 

427 


566 

5/ 

434 

861 

3 

56 

5 

9.35 481 

O'* 

9.36 624 

58 

0.63 376 

9.98 858 

3 

55 

6 

536 

55 

Pi4 

681 

57 

B7 

319 

855 

3 

54 

7 

590 

r.A 

738 

5/ 

By 

262 

852 

3 

53 

8 

644 

o*± 

795 

5/ 

By 

205 

849 

3 

52 

9 

698 

BA 

852 

D/ 

148 

846 

3 

51 

10 

9.35 752 

54 

9.36 909 

57 

0.63 091 

9.98 843 

3 

50 

11 

806 

54 

54 

9.36 966 

57 

0.63 034 

840 

3 

49 

12 

860 

B.A 

9.37 023 

57 

Pi7 

0.62 977 

837 

3 

48 

13 

914 

Oi 

080 

5/ 

Pi7 

920 

834 

3 

47 

14 

9.35 968 

ba 

137 

57 

863 

831 

3 

46 

15 

9.36 022 

0*4 

9.37 193 

56 

0.62 807 

9.98 828 

3 

45 

16 

075 

53 

BA 

250 

57 

750 

825 

3 

44 

17 

129 

04: 

pro 

306 

56 

694 

822 

3 

43 

18 

182 

0 O 

54 

363 

57 

CO 

637 

819 

3 

42 

19 

236 

CQ 

419 

50 

pr'r 

581 

816 

3 

41 

20 

9.36 289 

Do 

9.37 476 

57 

0.62 524 

9.98 813 

3 

40 

21 

342 

53 

CQ 

532 

56 

468 

810 

3 

39 

22 

395 

Do 

pr a 

588 

56 

412 

807 

3 

38 

23 

449 

54 

KQ 

644 

56 

cn 

356 

804 

3 

37 

24 

502 

Do 

KQ 

700 

50 

CO 

300 

801 

3 

36 

25 

9.36 555 

Do 

9.37 756 

50 

0.62 244 

9.98 798 

3 

35 

26 

608 

53 

(CQ 

812 

56 

188 

795 

3 

34 

27 

660 

oz 

CQ 

868 

56 

132 

792 

3 

33 

28 

713 

Do 

CQ 

924 

56 

076 

789 

3 

32 

29 

766 

06 

CQ 

9.37 980 

56 

0.62 020 

786 

3 

31 

30 

9.36 819 

Do 

9.38 035 

55 

0.61 965 

9.98 783 

3 

30 

31 

871 

52 

091 

56 

909 

780 

3 

29 

32 

924 

53 

147 

56 

853 

777 

3 

28 

33 

9.36 976 

52 

Cf) 

202 

55 

798 

774 

3 

27 

34 

9.37 028 

Du 

CQ 

257 

55 

743 

771 

3 

26 

35 

9.37 081 

Do 

9.38 313 

56 

0.61 687 

9.98 768 

3 

25 

36 

133 

52 

CQ 

368 

55 

632 

765 

3 

24 

37 

185 

0Z 

CQ 

423 

55 

577 

762 

3 

23 

38 

237 

OZ 

479 

56 

521 

759 

3 

22 

39 

289 

52 

CQ 

534 

55 

466 

756 

3 

21 

40 

9.37 341 

oz 

9.38 589 

55 

0.61 411 

9.98 753 

3 

20 

41 

393 

52 

CQ 

644 

55 

cc 

356 

750 

3 

19 

42 

445 

OZ 

BO 

699 

55 

pr pr 

301 

746 

4 

18 

43 

497 

OZ 

CQ 

754 

55 

pr a 

246 

743 

3 

17 

44 

549 

D Z 

C 1 

808 

54 

192 

740 

3 

16 

45 

9.37 600 

51 

9.38 863 

55 

0.61 137 

9.98 737 

3 

15 

46 

652 

52 

918 

55 

082 

734 

3 

14 

47 

703 

51 

9.38 972 

54 

0.61 028 

731 

3 

13 

48 

755 

52 

Cl 

9.39 027 

55 

ce 

0.60 973 

728 

3 

Q 

12 

49 

806 

01 

082 

55 

918 

725 

o 

Q 

11 

50 

9.37 858 

52 

9.39 136 

54 

0.60 864 

9.98 722 

5 

10 

51 

909 

51 

190 

54 

810 

719 

3 

9 

52 

9.37 960 

51 

245 

55 

755 

715 

4 

8 

53 

9.38 011 

51 

299 

54 

701 

712 

3 

7 

54 

062 

51 

353 

54 

647 

709 

3 

6 

55 

9.38 113 

51 

9.39 407 

54 

0.60 593 

9.98 706 

3 

5 

56 

164 

51 

461 

54 

539 

703 

3 

Q 

4 

57 

215 

51 

515 

54 

Pi/i 

485 

700 

o 

Q 

3 

58 

266 

DI 

Cl 

569 

54 

BA 

431 

697 

O 

Q 

2 

59 

317 

51 

pr 1 

623 

54 

BA 

377 

694 

O 

A 

1 

60 

9.38 368 

51 

9.39 677 

54 

0.60 323 

9.98 690 

4 

0 


L Cos* 

d 

L Cot* 

c d 

L Tan | 

L Sin* 

d 

/ 


76 ' 


[ 51 ] 



















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 14' 


L Sin* 


9.38 368 

418 
469 
519 
570 
9.38 620 

670 
721 
771 
821 
9.38 871 


921 

9.38 971 

9.39 021 
071 

9.39 121 

170 

220 

270 

Q1 Q 

9.39 369 


418 
467 
517 
566 
9.39 615 

664 
713 
762 
811 
9.39 860 


909 

9.39 958 

9.40 006 
055 

9.40 103 

152 
200 
249 
297 
9.40 346 


394 
442 
490 
538 
9.40 586 

634 
682 
730 
778 
9.40 825 


873 

Q91 

9.40 968 

9.41 016 
9.41 063 

111 
. 158 
205 
252 
9.41 300 


L Cos * 


L Tan* 


9.39 677 

731 
785 
838 
892 
9.39 945 

9.39 999 

9.40 052 
106 
159 

9.40 212 


266 
319 
372 
425 
9.40 478 

531 
584 
636 
689 
9.40 742 


795 

847 

900 

9.40 952 

9.41 005 

057 
109 
161 
214 
9.41 266 


318 
370 
422 
474 
9.41 526 

578 
629 
681 
733 
9.41 784 


836 

887 

939 

9.41 990 

9.42 041 

093 
144 
195 
246 
9.42 297 


348 
399 
450 
501 
9.42 552 

603 
653 
704 
755 
9.42 805 


c d 


L Cot 

L Cos* 

0.60 323 

9.98 690 

269 

687 

215 

684 

162 

681 

108 

678 

0.60 055 

9.98 675 

0.60 001 

671 

0.59 948 

668 

894 

665 

841 

662 

0.59 788 

9.98 659 

734 

656 

681 

652 

628 

649 

575 

646 

0.59 522 

9.98 643 

469 

640 

416 

636 

364 

633 

311 

630 

0.59 258 

9.98 627 

205 

623 

153 

620 

100 

617 

0.59 048 

614 

0.58 995 

9.98 610 

943 

607 

891 

604 

839 

601 

786 

597 

0.58 734 

9.98 594 

682 

591 

630 

588 

578 

584 

526 

581 

0.58 474 

9.98 578 

422 

574 

371 

571 

319 

568 

267 

565 

0.58 216 

9.98 561 

164 

558 

113 

555 

061 

551 

0.58 010 

548 

0.57 959 

9.98 545 

907 

541 

856 

538 

805 

535 

754 

531 

0.57 703 

9.98 528 

652 

525 

601 

521 

550 

518 

499 

515 

0.57 448 

9.98 511 

397 

508 

347 

505 

296 

501 

245 

498 

0.57 195 

9.98 494 

L Tan 

L Sin* 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with “*.” 



54 

53 

52 

1 

5.4 

5.3 

5.2 

2 

10.8 

10.6 

10.4 

3 

16.2 

15.9 

15.6 

4 

21.6 

21.2 

20.8 

5 

27.0 

26.5 

26.0 

6 

32.4 

31.8 

31.2 

7 

37.8 

37.1 

36.4 

8 

43.2 

42.4 

41.6 

9 

48.6 

47.7 

46.8 



51 

60 

49 

1 

5.1 

5 

4.9 

2 

10.2 

10 

9.8 

3 

15.3 

15 

14.7 

4 

20.4 

20 

19.6 

5 

25.5 

25 

24.5 

6 

30.6 

30 

29.4 

7 

35.7 

35 

34.3 

8 

40.8 

40 

39.2 

9 

45.9 

45 

44.1 



48 

47 

1 

4.8 

4.7 

2 

9.6 

9.4 

3 

14.4 

14.1 

4 

19.2 

18.8 

5 

24.0 

23.5 

6 

28.8 

28.2 

7 

33.6 

32.9 

8 

38.4 

37.6 

9 

43.2 

42.3 



4 

3 

1 

0.4 

0.3 

2 

0.8 

0.6 

3 

1.2 

0.9 

4 

1.6 

1.2 

5 

2.0 

1.5 

6 

2.4 

1.8 

7 

2.8 

2.1 

8 

3.2 

2.4 

9 

3.6 

2.7 


[ 52 ] 


75 ' 



















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 15 



74 < 


[ 53 ] 























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 16' 



[ 54 ] 


73 ' 
























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 17 c 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with 



45 

44 

43 

6 

1 

4.5 

4.4 

4.3 

7 

8 

2 

9.0 

8.8 

8.6 

9 

3 

13.5 

13.2 

12.9 

10 

4 

18.0 

17.6 

17.2 

5 

22.5 

22.0 

21.5 

11 

6 

27.0 

26.4 

25.8 

12 

7 

31.5 

30.8 

30.1 

13 

8 

36.0 

35.2 

34.4 

14 

9 

40.5 

39.6 

38.7 

15 



42 

41 

1 

4.2 

4.1 

2 

8.4 

8.2 

3 

12.6 

12.3 

4 

16.8 

16.4 

5 

21.0 

20.5 

6 

25.2 

24.6 

7 

29.4 

28.7 

8 

33.6 

32.8 

9 

37.8 

36.9 



40 

39 

1 

4 

3.9 

2 

8 

7.8 

3 

12 

11.7 

4 

16 

15.6 

5 

20 

19.5 

6 

24 

23.4 

7 

28 

27.3 

8 

32 

31.2 

9 

36 

35.1 



5 

4 

3 

1 

0.5 

0.4 

0.3 

2 

1.0 

0.8 

0.6 

3 

1.5 

1.2 

0.9 

4 

2.0 

1.6 

1.2 

5 

2.5 

2.0 

1.5 

6 

3.0 

2.4 

1.8 

7 

3.5 

2.8 

2.1 

8 

4.0 

3.2 

2.4 

9 

4.5 

3.6 

2.7 


L Sin* 


9.46 594 

635 
676 
717 
758 
9.46 800 

841 

882 

qoq 

9.46 964 

9.47 005 


045 
086 
127 
168 
9.47 209 

249 
290 
330 
371 
9.47 411 


452 
492 
533 
573 
9.47 613 

654 
694 
734 
774 
9.47 814 


854 

894 

934 

9.47 974 

9.48 014 

054 
094 
133 
173 
9.48 213 


252 
292 
332 
371 
9.48 411 

450 
490 
529 
568 
9.48 607 


647 
686 
725 
764 
9.48 803 

842 
881 
920 
959 
9.48 998 


L Cos* 


41 

41 

41 

41 

42 

41 

41 

41 

41 

41 

40 

41 
41 
41 
41 

40 

41 

40 

41 

40 

41 

40 

41 
40 

40 

41 
40 
40 
40 
40 

40 

40 

40 

40 

40 

40 

40 

39 

40 
40 

39 

40 
40 

39 

40 

39 

40 
39 
39 

39 

40 
39 
39 
39 
39 

39 

39 

39 

39 

39 


L Tan* 


9.48 534 

579 
624 
669 
714 
9.48 759 

804 
849 
894 
939 
9.48 984 


9.49 029 
073 
118 
163 
9.49 207 

252 
296 
341 
385 
9.49 430 


474 
519 
563 
607 
9.49 652 

696 
740 
784 
828 
9.49 872 


916 

9.49 960 

9.50 004 
048 

9.50 092 

136 
180 
223 
267 
9.50 311 


355 
398 
442 
485 
9.50 529 

572 
616 
659 
703 
9.50 746 


789 

833 

876 

Q1 Q 

9.50 962 

9.51 005 
048 
092 
135 

9.51 178 


c d 


45 

45 

45 

45 

45 

44 

45 
45 

44 

45 

44 

45 

44 

45 

44 

45 
44 

44 

45 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

44 

43 

44 
44 

44 

43 

44 

43 

44 

43 

44 

43 

44 
43 

43 

44 
43 
43 
43 

43 

43 

44 
43 
43 


1 L Cot 

L Cos* 

0.51 466 

9.98 060 

421 

056 

376 

052 

331 

048 

286 

044 

0.51 241 

9.98 040 

196 

036 

151 

032 

106 

029 

061 

025 

0.51 016 

9.98 021 

0.50 971 

017 

927 

013 

882 

009 

837 

005 

0.50 793 

9.98 001 

748 

9.97 997 

704 

993 

659 

989 

615 

986 

0.50 570 

9.97 982 

526 

978 

481 

974 

437 

970 

393 

966 

0.50 348 

9.97 962 

304 

958 

260 

954 

216 

950 

172 

946 

0.50 128 

9.97 942 

084 

938 

0.50 040 

934 

0.49 996 

930 

952 

926 

0.49 908 

9.97 922 

864 

918 

820 

914 

777 

910 

733 

906 

0.49 689 

9.97 902 

645 

898 

602 

894 

558 

890 

515 

886 

0.49 471 

9.97 882 

428 

878 

384 

874 

341 

870 

297 

866 

0.49 254 

9.97 861 

211 

857 

167 

853 

124 

849 

081 

845 

0.49 038 

9.97 841 

0.48 995 

837 

952 

833 

908 

829 

865 

825 

0.48 822 

9.97 821 

L Tan 

L Sin * 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


72 c 


[ 55 ] 


























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 18 


L Sin * 


L Tan * 

c d 

L Cot 

L Cos * 

9.51 178 


0.48 822 

9.97 821 

221 
264 
306 
349 
9.51 392 

43 

43 

42 

43 
43 

779 
736 
694 
651 
0.48 608 

817 
812 
808 
804 
9.97 800 

435 
478 
520 
563 
9.51 606 

43 

43 

42 

43 
43 

565 
522 
480 
437 
0.48 394 

796 
792 
788 
784 
9.97 779 

648 
691 
734 
776 
9.51 819 

42 

43 
43 

42 

43 

352 
309 
266 
224 
0.48 181 

775 
771 
767 
763 
9.97 759 

861 

903 

946 

9.51 988 

9.52 031 

42 

42 

43 

42 

43 

139 
097 
054 
0.48 012 
0.47 969 

754 
750 
746 
742 
9.97 738 

073 
115 
157 
200 
9.52 242 

42 

42 

42 

43 
42 

927 
885 
843 
800 
0.47 758 

734 
729 
725 
721 
9.97 717 

284 
326 
368 
410 
9.52 452 

42 

42 

42 

42 

42 

716 
674 
632 
590 
0.47 548 

713 
708 
704 
700 
9.97 696 

494 
536 
578 
620 
9.52 661 

42 

42 

42 

42 

41 

506 
464 
422 
380 
0.47 339 

691 
687 
683 
679 
9.97 674 

703 
745 
787 
829 
9.52 870 

42 

42 

42 

42 

41 

297 
255 
213 
171 
0.47 130 

670 
666 
662 
657 
9.97 653 

912 

953 

9.52 995 

9.53 037 
9.53 078 

42 

41 

42 
42 
41 

088 
047 
0.47 005 
0.46 963 
0.46 922 

649 
645 
640 
636 
9.97 632 

120 
161 
202 
244 
9.53 285 

42 

41 

41 

42 
41 

880 
839 
798 
756 
0.46 715 

628 
623 
619 
615 
9.97 610 

327 
368 
409 
450 
9.53 492 

42 

41 

41 

41 

42 

673 
632 
591 
550 
0.46 508 

606 
602 
597 
593 
9.97 589 

533 
574 
615 
656 
9.53 697 

41 

41 

41 

41 

41 

467 
426 
385 
344 
0.46 303 

584 
580 
576 
571 
9.97 567 

L Cot * 

c d 

L Tan 

L Sin * 


Prop . Parts 


9.48 998 

9.49 037 
076 
115 
153 

9.49 192 

231 
269 
308 
347 
9.49 385 


424 
462 
500 
539 
9.49 577 

615 
654 
692 
730 
9.49 768 


806 
844 
882 
920 
9.49 958 

9.49 996 

9.50 034 
072 
110 

9.50 148 


185 
223 
261 
298 
9.50 336 

374 
411 
449 
486 
9.50 523 


561 
598 
635 
673 
9.50 710 

747 
784 
821 
858 
9.50 896 


933 

9.50 970 

9.51 007 
043 

9.51 080 

117 
154 
191 
227 
9.51 264 


.L Cos * 



43 

42 

41 

1 

4.3 

4.2 

4.1 

2 

8.6 

8.4 

8.2 

3 

12.9 

12.6 

12.3 

4 

17.2 

16.8 

16.4 

5 

21.5 

21.0 

20.5 

6 

25.8 

25.2 

24.6 

7 

30.1 

29.4 

28.7 

8 

34.4 

33.6 

32.8 

9 

38.7 

37.8 

36.9 



39 

38 

1 

3.9 

3.8 

2 

7.8 

7.6 

3 

11.7 

11.4 

4 

15.6 

15.2 

5 

19.5 

19.0 

6 

23.4 

22.8 

7 

27.3 

26.6 

8 

31.2 

30.4 

9 

35.1 

34.2 



37 

36 

1 

3.7 

3.6 

2 

7.4 

7.2 

3 

11.1 

10.8 

4 

14.8 

14.4 

5 

18.5 

18.0 

6 

22.2 

21.6 

7 

25.9 

25.2 

8 

29.6 

28.8 

9 

33.3 

32.4 



5 

4 

1 

0.5 

0.4 

2 

1.0 

0.8 

3 

1.5 

1.2 

4 

2.0 

1.6 

5 

2.5 

2.0 

6 

3.0 

2.4 

7 

3.5 

2.8 

8 

4.0 

3.2 

9 

4.5 

3.6 


[ 56 ] 


71 ' 











































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 19' 



70 ‘ 


[ 57 ] 
































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 20° 


L Sin* 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9.53 405 

440 
475 
509 
544 
9.53 578 

613 
647 
682 
716 
9.53 751 


785 
819 
854 
888 
9.53 922 

957 

9.53 991 

9.54 025 
059 

9.54 093 


127 
161 
195 
229 
9.54 263 

297 
331 
365 
399 
9.54 433 


466 
500 
534 
567 
9.54 601 

635 
668 
702 
735 
9.54 769 


L Tan* 


802 
836 
869 
903 
9.54 936 

9.54 969 

9.55 003 
036 
069 

9.55 102 


136 
169 
202 
235 
9.55 268 

301 
334 
367 
400 
9.55 433 


35 

35 

34 

35 

34 

35 

34 

35 

34 

35 

34 

34 

35 
34 

34 

35 
34 
34 
34 
34 

34 

34 

34 

34 

34 

34 

34 

34 

34 

34 

33 

34 
34 

33 

34 

34 

33 

34 

33 

34 

33 

34 

33 

34 
33 

33 

34 
33 
33 

33 

34 
33 
33 
33 
33 

33 

33 

33 

33 

33 


9.56 107 

146 
185 
224 
264 
9.56 303 

342 

381 

420 

,ikq 

9.56 498 


537 
576 
615 
654 
9.56 693 

732 
771 
810 
849 
9.56 887 


926 

9.56 965 

9.57 004 

. 042 

9.57 081 

120 
158 
197 
235 
9.57 274 


312 
351 
389 
428 
9.57 466 

504 
543 
581 
619 
9.57 658 


696 

734 


L Cos* 


L Cot* 


c d 

L Cot 

L Cos* 

d 



0.43 893 

9.97 299 


60 

39 

854 

294 

5 

59 

39 

815 

289 

5 

58 

39 

776 

285 

4 

57 

40 

736 

280 

5 

56 

39 

0.43 697 

9.97 276 

4 

55 

39 

658 

271 

5 

54 

39 

619 

266 

5 

A 

53 

39 

580 

262 

4 

er 

52 

39 

541 

257 

O 

e 

51 

39 

0.43 502 

9.97 252 

o 

50 

39 

463 

248 

4 

49 

39 

424 

.243 

5 

48 

39 

385 

238 

5 

47 

39 

346 

234 

4 

46 

39 

0.43 307 

9.97 229 

5 

45 

39 

268 

224 

5 

44 

39 

229 

220 

4 

43 

39 

190 

215 

5 

42 

39 

151 

210 

5 

A 

41 

38 

0.43 113 

9.97 206 

4 

40 

39 

074 

201 

5 

39 

39 

0.43 035 

196 

5 

A 

38 

39 

0.42 996 

192 

4 

er 

37 

38 

958 

187 

o 

c 

36 

39 

0.42 919 

9.97 182 

O 

35 

39 

880 

178 

4 

34 

38 

842 

173 

5 

33 

39 

803 

168 

5 

32 

38 

765 

163 

5 

31 

39 

0.42 726 

9.97 159 

4 

30 

" 38 

688 

154 

5 

29 

39 

649 

149 

5 

28 

38 

611 

145 

4 

27 

39 

572 

140 

5 

26 

38 

0.42 534 

9.97 135 

5 

25 

38 

496 

130 

5 

24 

39 

457 

126 

4 

23 

38 

419 

121 

5 

22 

38 

381 

116 

5 

21 

39 

0.42 342 

9.97 111 

5 

20 

38 

304 

107 

4 

19 

38 

266 

102 

5 

18 

38 

228 

097 

5 

17 

i 38 

190 

092 

5 

16 

i 39 

0.42 151 

9.97 087 

5 

15 

38 

113 

083 

4 

14 

38 

1 o n 

075 

078 

5 

13 

38 

1 OO 

0.42 037 

073 

5 

12 

38 

0.41 999 

068 

o 

c 

11 

i 38 

0.41 961 

9.97 063 

O 

10 

r 38 

923 

059 

4 

K 

9 

• 38 

885 

054 

o 

er 

8 

! 38 

847 

049 

0 

7 

, 38 

809 

044 

5 

c 

6 

j 38 

0.41 771 

9.97 039 

O 

5 

r 38 

733 

035 

4 

4 

t 11 

696 

030 

5 

e 

3 

> 38 

658 

025 

o 

E 

2 

1 38 

620 

020 

o 

e 

1 

^ 38 

0.41 582 

9.97 015 

O 

0 

c d 

L Tan 

L Sin* 

d 

' 


Prop. Parts 


Subtract 10 from each en- 



40 

39 

38 

1 

4 

3.9 

3.8 

2 

8 

7.8 

7.6 

3 

12 

11.7 

11.4 

4 

16 

15.6 

15.2 

5 

20 

19.5 

19.0 

6 

24 

23.4 

22.8 

7 

28 

27.3 

26.6 

8 

32 

31.2 

30.4 

9 

36 

35.1 

34.2 



37 

35 

1 

3.7 

3.5 

2 

7.4 

7.0 

3 

11.1 

10.5 

4 

14.8 

14.0 

5 

18.5 

17.5 

6 

22.2 

21.0 

7 

25.9 

24.5 

8 

29.6 

28.0 

9 

33.3 

31.5 



34 

33 

1 

3.4 

3.3 

2 

6.8 

6.6 

3 

10.2 

9.9 

4 

13.6 

13.2 

5 

17.0 

16.5 

6 

20.4 

19.8 

7 

23.8 

23.1 

8 

27.2 

26.4 

9 

30.6 

29.7 



5 

4 

1 

0.5 

0.4 

2 

1.0 

0.8 

3 

1.5 

1.2 

4 

2.0 

1.6 

5 

2.5 

2.0 

6 

3.0 

2.4 

7 

3.5 

2.8 

8 

4.0 

3.2 

9 

4.5 

3.6 


69 ° 


[ 58 ] 






























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 21' 



68' 


[59] 













































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 22 



[ 60 ] 


67 ' 













































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 23 



66' 


[61] 



























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 24 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


9.60 931 

960 

9.60 988 

9.61 016 
045 

9.61 073 

101 
129 
158 
186 
9.61 214 


242 
270 
298 
326 
9.61 354 

382 
411 
438 
466 
9.61 494 


522 
550 
578 
606 
9.61 634 

662 
689 
717 
745 
9.61 773 


800 
828 
856 
883 
9.61 911 

939 

966 

9.61 994 

9.62 021 
9.62 049 


076 

104 

131 

1 

9.62 186 

214 
241 
268 
296 
9.62 323 


350 
377 
405 
432 
9.62 459 

486 
513 
541 
568 
9.62 595 


L Cos* 


9.64 858 

892 

926 

960 

9.64 994 

9.65 028 

062 
096 
130 
164 
9.65 197 


231 
265 
299 
333 
9.65 366 

400 
434 
467 
501 
9.65 535 


568 
602 
636 
669 
9.65 703 

736 

770 

803 

9.65 Ifo 


904 

937 

9.65 971 

9.66 004 
9.66 038 

071 
104 
138 
171 
9.66 204 


238 
271 
304 
337 
9.66 371 

404 
437 
470 
503 
9.66 537 


570 
603 
636 
669 
9.66 702 

735 

768 

801 

QQJ. 

9.66 867 


L Cot* 


c d 


L Cot 

L Cos* 

0.35 142 

9.96 073 

108 

067 

074 

062 

040 

056 

0.35 006 

050 

0.34 972 

9.96 045 

938 

039 

904 

034 

870 

028 

836 

022 

0.34 803 

9.96 017 

769 

011 

735 

005 

701 

9.96 000 

667 

9.95 994 

0.34 634 

9.95 988 

600 

982 

566 

977 

533 

971 

499 

965 

0.34 465 

9.95 960 

432 

954 

398 

948 

364 

942 

331 

937 

0.34 297 

9.95 931 

264 

925 

230 

920 

197 

914 

163 

908 

0.34 130 

9.95 902 

096 

897 

063 

891 

0.34 029 

885 

0.33 996 

879 

0.33 962 

9.95 873 

929 

868 

896 

862 

862 

856 

829 

850 

0.33 796 

9.95 844 

762 

839 

729 

833 

696 

827 

663 

821 

0.33 629 

9.95 815 

596 

810 

563 

804 

530 

798 

497 

792 

0.33 463 

9.95 786 

430 

780 

397 

775 

364 

769 

331 

763 

0.33 298 

9.95 757 

265 

751 

232 

745 

199 

739 

166 

733 

0.33 133 

9.95 728 

L Tan 

L Sin* 


d 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
unth “*.” 



34 

33 

1 

3.4 

3.3 

2 

6.8 

6.6 

3 

10.2 

9.9 

4 

13 . 6 - 

13.2 

5 

17.0 

16.5 

6 

20.4 

19.8 

7 

23.8 

23.1 

8 

27.2 

26.4 

9 

30.6 

29.7 



29 

28 

27 

1 

2.9 

2.8 

2.7 

2 

5.8 

5.6 

5.4 

3 

8.7 

8.4 

8.1 

4 

11.6 

11.2 

10.8 

5 

14.5 

14.0 

13.5 

6 

17.4 

16.8 

16.2 

7 

20.3 

19.6 

18.9 

8 

23.2 

22.4 

21.6 

9 

26.1 

25.2 

24.3 



6 

6 

1 

0.6 

0.5 

2 

1.2 

1.0 

3 

1.8 

1.5 

4 

2.4 

2.0 

5 

3.0 

2.5 

6 

3.6 

3.0 

7 

4.2 

3.5 

8 

4.8 

4.0 

9 

5.4 

4.5 


[ 62 ] 


65 



















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 25' 



64' 


[ 63 ] 















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS —26 



[ 64 ] 


63' 








































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 27 



62 


[ 65 ] 























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 28 


/ 

L Sin* 

d 

L Tan* 

c d 

L Cot 

L Cos* 

d 


0 

9.67 161 


9.72 567 


0.27 433 

9.94 593 


60 

1 

185 

24 

598 

31 

402 

587 

6 

7 

59 

2 

208 

23 

628 

30 

372 

580 

4 

58 

3 

232 

24 

659 

31 

341 

573 

7 

57 

4 

256 

24 

689 

30 

311 

567 

6 

56 

5 

9.67 280 

24 

9.72 720 

31 

0.27 280 

9.94 560 

7 

55 

6 

303 

23 

750 

30 

250 

553 

7 

7 

54 

7 

327 

24 

780 

30 

220 

546 

4 

53 

8 

350 

23 

811 

31 

189 

540 

o 

7 

52 

9 

374 

24 

841 

30 

159 

533 

4 

7 

51 

10 

9.67 398 

24 

9.72 872 

31 

0.27 128 

9.94 526 

4 

50 

11 

421 

23 

902 

30 

098 

519 

7 

a 

49 

12 

445 

24 

932 

30 

068 

513 

O 

7 

48 

13 

468 

23 

963 

31 

037 

506 

4 

7 

47 

14 

492 

24 

9.72 993 

30 

0.27 007 

499 

4 

7 

46 

15 

9.67 515 

23 

9.73 023 

30 

0.26 977 

9.94 492 

4 

45 

16 

539 

24 

054 

31 

946 

485 

7 

44 

17 

562 

23 

084 

30 

916 

479 

6 

7 

43 

18 

586 

24 

114 

30 

886 

472 

4 

7 

42 

19 

609 

23 

144 

30 

856 

465 

7 

7 

41 

20 

9.67 633 

24 

9.73 175 

31 

0.26 825 

9.94 458 

4 

40 

21 

656 

23 

205 

30 

795 

451 

7 

a 

39 

22 

680 

24 

235 

30 

765 

445 

D 

7 

38 

23 

703 

23 

265 

30 

735 

438 

4 

7 

37 

24 

726 

23 

295 

30 

705 

431 

4 

7 

36 

25 

9.67 750 


9.73 326 

31 

0.26 674 

9.94 424 

4 

35 

26 

773 

23 

356 

30 

644 

417 

7 

34 

27 

796 

23 

386 

30 

614 

410 

7 

c 

33 

28 

820 

24 

416 

30 

584 

404 

O 

7 

32 

29 

843 

23 

446 

30 

554 

397 

4 

7 

31 

30 

9.67 866 

23 

9.73 476 

30 

0.26 524 

9.94 390 

4 

30 

31 

890 

24 

507 

31 

493 

383 

7 

29 

32 

913 

23 

537 

30 

463 

376 

7 

7 

28 

33 

936 

23 

567 

30 

433 

369 

4 

7 

27 

34 

959 

23 

597 

30 

403 

362 

4 

7 

26 

35 

9.67 982 

23 

9.73 627 

30 

0.26 373 

9.94 355 

7 

25 

36 

9.68 006 

24 

657 

30 

343 

349 

6 

24 

37 

029 

23 

687 

30 

313 

342 

7 

7 

23 

38 

052 

23 

717 

30 

283 

335 

4 

7 

22 

39 

075 

23 

no 

747 

30 

on , 

253 

328 

4 

7 

21 

40 

9.68 098 

16 

9.73 777 

oU ! 

0.26 223 

9.94 321 

4 

20 

41 

121 

23 

807 

30 

193 

314 

7 

19 

42 

144 

23 

837 

30 

163 

307 

7 

7 

18 

43 

167 

23 

867 

30 

133 

300 

4 

7 

17 

44 

190 

23 

897 

30 

103 

293 

4 

7 

16 

45 

9.68 213 

23 

9.73 927 

30 

0.26 073 

9.94 286 

7 

15 

46 

237 

24 

no 

957 

30 

on 

043 

279 

7 

a 

14 

47 

260 

16 

no 

9.73 987 

ou 

on 

0.26 013 

273 

O 

7 

13 

48 

283 

16 

9.74 017 

6U 

on 

0.25 983 

266 

4 

7 

12 

49 

305 

22 

no 

047 

6U 

on 

953 

259 

4 

7 

11 

50 

9.68 328 

16 

9.74 077 

6U 

0.25 923 

9.94 252 

4 

10 

51 

351 

23 

no 

107 

30 

893 

245 

7 

7 

9 

52 

374 

16 

137 

30 

863 

238 

4 

7 

8 

53 

397 

23 

no 

166 

29 

on 

834 

231 

4 

7 

7 

54 

420 

16 

no 

196 

oU 

QA 

804 

224 

4 

7 

6 

55 

9.68 443 

16 

9.74 226 

oU 

0.25 774 

9.94 217 

4 

5 

56 

466 

23 

256 

30 

on 

744 

210 

7 

7 

4 

57 

489 

23 

286 

oU 

714 

203 

4 

7 

3 

58 

512 

23 

316 

30 

684 

196 

4 

7 

2 

59 

534 

22 

no 

345 

29 

QA 

655 

189 

4 

7 

1 

60 

9.68 557 

16 

9.74 375 

oU 

0.25 625 

9.94 182 

4 

0 


L Cos* 

d 

L Cot* 

c d 

L Tan 

L Sin * 

d 

/ 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked 
with “ 



31 

30 

29 

1 

3.1 

3 

2.9 

2 

6.2 

6 

5.8 

3 

9.3 

9 

8.7 

4 

12.4 

12 

11.6 

5 

15.5 

15 

14.5 

6 

18.6 

18 

17.4 

7 

21.7 

21 

20.3 

8 

24.8 

24 

23.2 

9 

27.9 

27 

26.1 



24 

23 

22 

1 

2.4 

2.3 

2.2 

2 

4.8 

4.6 

4.4 

3 

7.2 

6.9 

6.6 

4 

9.6 

9.2 

8.8 

5 

12.0 

11.5 

11.0 

6 

14.4 

13.8 

13.2 

7 

16.8 

16.1 

15.4 

8 

19.2 

18.4 

17.6 

9 

21.6 

20.7 

19.8 



7 

6 

1 

0.7 

0.6 

2 

1.4 

1.2 

S 

2.1 

1.8 

4 

2.8 

2.4 

5 

3.5 

3.0 

6 

4.2 

3.6 

7 

4.9 

4.2 

8 

5.6 

4.8 

9 

6.3 

5.4 


[ 66 ] 


6L 

















































60' 


[ 67 ] 















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 30' 



[ 68 ] 


59' 
















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 31° 


Prop. Parts 

/ 

L Sin* 

d 

L Tan* 

c d 

L Cot 

L Cos* 

d 


Subtract 10 from each en- 

0 

9.71 

184 

21 

21 

21 

21 

9.77 877 

29 

29 

28 

29 

0.22 123 

9.93 307 


60 

try in the columns marked 
with 

1 

2 

3 

4 

9.71 

205 

226 

247 

268 

906 
935 
963 
9.77 992 

094 

065 

037 

0.22 008 


299 

291 

284 

27 fi 

8 

8 

7 

8 

59 

58 

57 

56 




5 

289 


9.78 

; 020 

28 

0.21 980 

9.93 269 

7 

55 




6 

7 


310 

331 

21 

21 

21 


049 

077 

29 

28 

29 

951 

923 


261 

253 

8 

8 

54 

53 


29 

28 

8 

9 

10 

9.71 

352 

373 

393 

21 

20 

9.78 

106 
135 
i 163 

29 

28 

894 
865 
0.21 837 

246 
238 
9.93 230 

7 

8 

8 

52 

51 

50 

1 

2 

3 

4 

5 

2.9 

5.8 

8.7 

11.6 

14.5 

2.8 

5.6 

8.4 

11.2 

14.0 

11 

12 

13 

14 

15 

9.71 

414 

435 

456 

477 

498 

21 

21 

21 

21 

21 

9.78 

192 

220 

249 

277 

306 

29 

28 

29 

28 

29 

808 
780 
751 
723 
0.21 694 

223 
215 
207 
200 
9.93 192 

' 7 

8 

8 

7 

8 

49 

48 

47 

46 

45 

6 

7 

8 

9 

17.4 

20.3 

23.2 

26.1 

16.8 

19.6 

22.4 

25.2 

16 

17 

18 

19 

20 

9.71 

519 

539 

560 

581 

602 

21 

20 

21 

21 

21 

9.78 

334 

363 

391 

419 

448 

28 

29 

28 

28 

29 

666 
637 
609 
581 
0.21 552 

184 
177 
169 
161 
9.93 154 

8 

7 

8 

8 

7 

44 

43 

42 

41 

40 




21 


622 

20 

21 


476 

28 

9Q 

524 


146 

8 

39 




22 


643 

21 


505 

28 

495 


138 

8 

38 




23 


664 

21 


533 

9Q 

467 


131 

7 

37 




24 

25 

9.71 

685 

705 

20 

9.78 

562 

590 

28 

438 
0.21 410 

9.93 

123 

; ns 

8 

8 

36 

35 




26 


726 

21 

21 


618 

28 

90 

382 


108 

7 

34 


21 

20 

27 

28 


747 

767 

20 


647 

675 

28 

353 

325 


100 

092 

8 

8 

33 

32 

1 

2.1 

2 

29 

9.71 

788 

21 

21 


704 

29 

9ft 

296 


084 

8 

31 

2 

4.2 

4 

30 

809 

9.78 

732 

&o 

0.21 268 

9.93 

077 

7 

30 

3 

4 

5 

6 

7 

8 

6.3 

8.4 

10.5 

12.6 

14.7 

16.8 

6 

8 

10 

12 

14 

16 

31 

32 

33 

34 

35 

9.71 

829 

850 

870 

891 

911 

20 

21 

20 

21 

20 

9.78 

760 

789 

817 

845 

874 

28 

29 

28 

28 

29 

240 

211 

183 

155 

0.21 126 

9.93 

069 

061 

053 

046 

038 

8 

8 

8 

7 

8 

29 

28 

27 

26 

25 

9 

18.9 

18 

36 


932 

21 

20 


902 

28 

28 

098 


030 

8 

24 




37 


952 

21 


930 

29 

070 


022 

8 

23 




38 

9.71 

973 

21 


959 

9R 

041 


014 

8 

22 




39 

994 

20 

9.78 

987 

£ o 

9ft 

0.21 013 

9.93 

007 

7 

21 




40 

9.72 

014 

9.79 

015 

40 

0.20 985 

9.92 

999 

8 

20 




41 


034 

20 

21 


043 

28 

9Q 

957 


991 

8 

19 




42 


055 

20 


072 

9ft 

928 


983 

8 

18 




43 


075 

21 


100 

4 o 

9ft 

900 


976 

7 

17 




44 

9.72 

096 

20 


128 

40 

28 

90 

872 


968 

8 

16 


8 

7 

45 

116 

21 

9.79 

156 

0.20 844 

9.92 

960 

8 

15 

1 

2 

3 

0.8 

1.6 

2.4 

0.7 

1.4 

2.1 

46 

47 

48 


137 

157 

177 

20 

20 

21 


185 

213 

241 

28 

28 

28 

815 

787 

759 


952 

944 

936 

8 

8 

8 

14 

13 

12 

4 

5 

3.2 

4.0 

2.8 

3.5 

49 

50 

9.72 

198 

218 

20 

9.79 

269 

297 

28 

731 
0.20 703 

9.92 

929 

921 

8 

11 

10 

6 

4.8 

4.2 

51 


238 

20 

91 

* 

326 

29 

674 


913 

8 

9 

7 

5.6 

4.9 

52 


259 

90 


354 

28 

646 


905 

8 

8 

8 

6.4 

5.6 

53 


279 

20 


382 

28 

9ft 

618 


897 

8 

7 

9 

7.2 

6.3 

54 


299 

21 


410 

Zo 

9ft 

590 


889 

8 

6 




55 

9.72 

320 

9.79 

438 

ZO 

0.20 562 

9.92 

881 

8 

5 




56 


340 

20 


466 

28 

534 


874 

7 

4 




57 


360 

20 

91 


495 

29 

505 


866 

8 

3 




58 


381 

zl 

90 


523 

28 

477 


858 

8 

2 




59 


401 

ZYJ 

on 


551 

28 

449 


850 

8 

1 




60 

9.72 

421 

ZU 

9.79 

579 

28 

0.20 421 

9.92 

842 

8 

0 





L Cos* 

d 

L Cot * 

c d 

L Tan 

L Sin* 

d 



58' 


[ 69 ] 































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS —32 



[ 70 ] 


57' 


















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 33 



56' 


[ 71 ] 


















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 34 



[ 72 ] 


55‘ 










































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 35° 



[ 73 ] 





























































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 36 


L Sin* 


L Cot 

L Cos* 

0.13 874 

9.90 796 

847 

787 

821 

777 

794 

768 

768 

759 

0.13 741 

9.90 750 

715 

741 

688 

731 

662 

722 

635 

713 

0.13 608 

9.90 704 

582 

694 

555 

685 

529 

676 

502 

667 

0.13 476 

9.90 657 

449 

648 

423 

639 

397 

630 

370 

620 

0.13 344 

9.90 611 

317 

602 

291 

592 

264 

583 

238 

574 

0.13 211 

9.90 565 

185 

555 

158 

546 

132 

537 

106 

527 

0.13 079 

9.90 518 

053 

509 

026 

499 

0.13 000 

490 

0.12 973 

480 

0.12 947 

9.90 471 

921 

462 

894 

452 

868 

443 

842 

434 

0.12 815 

9.90 424 

789 

415 

762 

405 

736 

396 

710 

386 

0.12 683 

9.90 377 

657 

368 

631 

358 

604 

349 

578 

339 

0.12 552 

9.90 330 

525 

320 

499 

311 

473 

301 

446 

292 

0.12 420 

9.90 282 

394 

273 

367 

263 

341 

254 

315 

244 

0.12 289 

9.90 235 

L Tan 

L Sin* 


Prop. Parts 


9.76 922 

939 

957 

974 

9.76 991 

9.77 009 

026 
043 
061 
078 

9.77 095 


112 
130 
147 
164 

9.77 181 

199 
216 
233 
250 

9.77 268 


285 
302 
319 
336 

9.77 353 

370 
387 
405 
422 

9.77 439 


456 
473 
490 
507 

9.77 524 

541 
558 
575 
592 

9.77 609 


626 
643 
660 
677 

9.77 694 

711 
728 
744 
761 

9.77 778 


795 
812 
829 
846 

9.77 862 

879 
896 
913 
930 

9.77 946 


9.86 126 

153 
179 
206 
232 

9.86 259 

285 
312 
338 
365 

9.86 392 


418 
445 
471 
498 

9.86 524 

551 
577 
603 
630 

9.86 656 


683 
709 
736 
762 

9.86 789 

815 
842 
868 
894 

9.86 921 


947 

9.86 974 

9.87 000 
027 

9.87 053 

079 
106 
132 
158 

9.87 185 


211 
238 
264 
290 

9.87 317 

343 
369 
396 
422 

9.87 448 


475 
501 
527 
554 

9.87 580 

606 
633 
659 
685 

9.87 711 


Subtract 10 from each en¬ 
try in the columns marked 
with “ *.” 



27 

26 

1 

2.7 

2.6 

2 

5.4 

5.2 

3 

8.1 

7.8 

4 

10.8 

10.4 

6 

13.5 

13.0 

6 

16.2 

15.6 

7 

18.9 

18.2 

8 

21.6 

20.8 

9 

24.3 

23.4 



18 

17 

16 

1 

1.8 

1.7 

1.6 

2 

3.6 

3.4 

3.2 

3 

5.4 

5.1 

4.8 

4 

7.2 

6.8 

6.4 

5 

9.0 

8.5 

8.0 

6 

10.8 

10.2 

9.6 

7 

12.6 

11.9 

11.2 

8 

14.4 

13.6 

12.8 

9 

16.2 

15.3 

14.4 



10 

9 

1 

1.0 

0.9 

2 

2.0 

1.8 

3 

3.0 

2.7 

4 

4.0 

3.6 

5 

5.0 

4.5 

6 

6.0 

5.4 

7 

7.0 

6.3 

8 

8.0 

7.2 

9 

9.0 

8.1 


L Cos* 


L Cot* 


c d 


[74] 


53' 








































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 37 6 


Prop. Parts 

/ 

L Sin* 

d 

L Tan * 

c d 

l L Cot 

L Cos* 

d 





0 

9.77 946 

17 

17 

17 

16 

17 

9.87 711 


0.12 289 

9.90 235 


60 




1 

2 

3 

4 

5 

963 

980 

9.77 997 

9.78 013 
9.78 030 

738 
764 
790 
817 
9.87 843 

27 

26 

26 

27 

26 

262 
236 
210 
183 
0.12 157 

225 
216 
206 
197 
9.90 187 

10 

9 

10 

9 

10 

59 

58 

57 

56 

55 


27 

26 

6 

7 

8 
9 

10 

047 
063 
080 
097 
9.78 113 

17 

16 

17 

17 

16 

869 
895 
922 
948 
9.87 974 

26 

26 

27 

26 

26 

131 

105 

078 

052 

0.12 026 

178 
168 
159 
149 
9.90 139 

9 

10 

9 

10 

10 

54 

53 

52 

51 

50 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2.7 

5.4 

8.1 

10.8 

13.5 
16.2 
18.9 

21.6 
24.3 

2.6 

5.2 

7.8 

10.4 

13.0 

15.6 

18.2 

20.8 

23.4 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

130 
147 
163 
180 
9.78 197 

213 
230 
246 
263 
9.78 280 

17 

17 

16 

17 

17 

16 

17 

16 

17 

17 

9.88 000 
027 
053 
079 
9.88 105 

131 
158 
184 
210 
9.88 236 

26 

27 

26 

26 

26 

26 

27 

26 

26 

26 

0.12 000 
0.11 973 
947 
921 
0.11 895 

869 
842 
816 
790 
0.11 764 

130 
120 
111 
101 
9.90 091 

082 
072 
063 
053 
9.90 043 

9 

10 

9 

10 

10 

9 

10 

9 

10 

10 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 




21 

22 

23 

24 

25 

296 
313 
329 
346 
9.78 362 

16 

17 

16 

17 

16 

262 
289 
315 
341 
9.88 367 

26 

27 

26 

26 

26 

738 
711 
685 
659 
0.11 633 

034 
024 
014 
9.90 005 
9.89 995 

9 

10 

10 

9 

10 

39 

38 

37 

36 

35 


17 

16 

26 

27 

28 

379 

395 

412 

17 

16 

17 

393 
420 
446 
472 
9.88 498 

26 

27 

26 

607 
580 
554 
528 
0.11 502 

985 

976 

QAA 

10 

9 

10 

34 

33 

QO 

1 

2 

1.7 

3.4 

1.6 

3.2 

29 

30 

428 
9.78 445 

16 

17 

26 

26 

yoo 
956 
9.89 947 

10 

9 

04 

31 

30 

3 

4 

5 

6 

7 

8 

5.1 

6.8 

8.5 

10.2 

11.9 

13.6 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

31 

32 

33 

34 

35 

461 
478 
494 
510 
9.78 527 

16 

17 

16 

16 

17 

524 
550 
577 
603 
9.88 629 

26 

26 

27 

26 

26 

476 
450 
423 
397 
0.11 371 

937 
927 
918 
908 
9.89 898 

10 

10 

9 

10 

10 

29 

28 

27 

26 

25 

9 

15.3 

14.4 

36 

37 

38 

39 

40 

543 
560 
576 
592 
9.78 609 

16 

17 

16 

16 

17 

655 
681 
707 
733 
9.88 759 

26 

26 

26 

26 

26 

345 
319 
293 
267 
0.11 241 

888 
879 
869 
859 
9.89 849 

10 

9 

10 

10 

10 

24 

23 

22 

21 

20 


10 

9 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

625 
642 
658 
674 
9.78 691 

707 
723 
739 
756 
9.78 772 

16 

17 

16 

16 

17 

16 

16 

16 

17 

16 

786 
812 
838 
864 
9.88 890 

27 

26 

26 

26 

26 

26 

26 

26 

26 

26 

214 

188 

162 

136 

0.11 110 

840 
830 
820 
810 
9.89 801 

9 

10 

10 

10 

9 

19 

18 

17 

16 

15 

1 

2 

3 

4 

5 

1.0 

2.0 

3.0 

4.0 

5.0 

0.9 

1.8 

2.7 

3.6 

4.5 

916 

942 

968 

9.88 994 

9.89 020 

084 

058 

032 

0.11 006 
0.10 980 

791 
781 
771 
761 
9.89 752 

10 

10 

10 

10 

9 

14 

13 

12 

11 

10 

6 

7 

8 

9 

6.0 

7.0 

8.0 

9.0 

5.4 

6.3 

7.2 

8.1 

51 

52 

53 

54 

55 

788 
805 
821 
837 
9.78 853 

16 

17 

16 

16 

16 

046 
073 
099 
125 
9.89 151 

26 

27 

26 

26 

26 

954 
927 
901 
875 
0.10 849 

742 
732 
722 
712 
9.89 702 

10 

10 

10 

10 

10 

9 

8 

7 

6 

5 




56 

57 

58 

59 

60 

869 
886 
902 
918 
9.78 934 

16 

17 

16 

16 

16 

177 
203 
229 
255 
9.89 281 

26 

26 

26 

26 

26 

823 

797 

771 

745 

0.10 719 1 

693 
683 
673 
663 
9.89 653 

9 

10 

10 

10 

10 

4 

3 

2 

1 

0 





L Cos* 

d 

L Cot * c 

: d 

L Tfln 

L Sin* 

d | r 


52’ 


[75] 





































































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 38' 



[76] 


51' 











































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 39' 



50‘ 


[77] 

















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 40' 


f 

L Sin* 

d 

L Tan* 

c d 

L Cot 

L Cos* 

d 


0 

9.80 807 


9.92 381 


0.07 619 

9.88 425 


60 

1 

822 

15 

407 

26 

593 

415 

10 

59 

2 

837 


433 

26 
O K 

567 

404 

11 

58 

3 

852 


458 

ZD 

542 

394 

10 

57 

4 

867 


484 

26 

516 

383 

11 

56 

5 

9.80 882 


9.92 510 

26 

0.07 490 

9.88 372 

11 

55 

6 

897 

15 

535 

25 

465 

362 

10 

54 

7 

912 


561 

26 

439 

351 

11 

53 

8 

927 


587 

26 

413 

340 

11 

52 

9 

942 


612 

25 

388 

330 

10 

51 

10 

9.80 957 


9.92 638 

26 

0.07 362 

9.88 319 

11 

50 

11 

972 

15 

663 

25 

337 

308 

11 

49 

12 

9.80 987 


689 

26 

311 

298 

10 

48 

13 

9.81 002 


715 

26 

285 

287 

11 

47 

14 

017 


740 

25 

260 

276 

11 

46 

15 

9.81 032 


9.92 766 

26 

0.07 234 

9.88 266 

10 

45 

16 

047 

15 

792 

26 

208 

255 

11 

44 

17 

061 


817 

25 

Oft 

183 

244 

11 

43 

18 

076 


843 

ZD 

O K 

157 

234 

10 

42 

19 

091 


868 

ZD 

132 

223 

11 

41 

20 

9.81 106 

TO 

9.92 894 

26 

0.07 106 

9.88 212 

11 

40 

21 

121 

15 

920 

26 

QC 

080 

201 

11 

39 

22 

136 


945 

ZD, 

Oft 

055 

191 

10 

38 

23 

151 


971 

ZD 

OK 

029 

180 

11 

37 

24 

166 


9.92 996 

ZD | 

Oft 

0.07 004 

169 

11 

36 

25 

9.81 180 


9.93 022 

ZD 

0.06 978 

9.88 158 

11 

35 

26 

195 

15 

048 

26 

O K 

952 

148 

10 

34 

27 

210 


073 

ZD 

9ft 

927 

137 

11 

33 

28 

225 

lu 

1 K 

099 

zo 

OK 

901 

126 

11 

32 

29 

240 


124 

ZD 

Oft 

876 

115 

11 

31 

30 

9.81 254 

1“± 

9.93 150 

ZD 

0.06 850 

9.88 105 

10 

30 

31 

269 

15 

15 

175 

25 

Oft 

825 

094 

11 

1 1 

29 

32 

284 

15 

201 

zu 

9ft 

799 

083 

11 

1 1 

28 

33 

299 

1 K 

227 

zu 

OK 

773 

072 

11 

27 

34 

314 

id 

1 4. 

252 

40 

9ft 

748 

061 

11 

26 

35 

9.81 328 


9.93 278 

ZD 

0.06 722 

9.88 051 

10 

25 

36 

343 

15 

15 

303 

25 

26 

697 

040 

11 

1 1 

24 

37 

358 

14 

329 

OK 

671 

029 

11 

1 1 

23 

38 

372 

15 

354 

40 

9 ft 

646 

018 

11 

1 1 

22 

39 

387 

15 

380 

zo 

Oft 

620 

9.88 007 

11 

1 1 

21 

40 

9.81 402 


9.93 406 

zo 

0.06 594 

9.87 996 

11 

20 

41 

417 

15 

14 

431 

25 

9fi 

569 

985 

11 

1 ft 

19 

42 

431 

15 

457 

zo 

OK 

543 

975 

1U 

1 1 

18 

43 

446 

15 

482 

40 

9ft 

518 

964 

11 

1 1 

17 

44 

461 

14 

508 

zo 

25 

492 

953 

11 

1 1 

16 

45 

9.81 475 


9.93 533 


0.06 467 

9.87 942 

11 

15 

46 

490 

15 

15 

559 

26 

25 

441 

931 

11 

1 1 

14 

47 

505 

14 

584 

26 

416 

920 

11 

1 1 

13 

48 

519 

15 

610 

26 

390 

909 

11 

1 1 

12 

49 

534 

15 

636 

25 

364 

898 

11 

11 

50 

9.81 549 


9.93 661 


0.06 339 

9.87 887 

11 

10 

51 

563 

14 

15 

687 

26 ‘ 
25 

313 

877 

10 

1 1 

9 

52 

578 

14 

712 

26 

288 

866 

11 

1 1 

8 

53 

592 

15 

738 

25 

262 

855 

11 

7 

54 

607 

15 

763 

26 

237 

844 

11 

1 1 

6 

55 

9.81 622 


9.93 789 


0.06 211 

9.87 833 

11 

5 

56 

636 

14 

814 

25 

186 

822 

11 

4 

57 

651 

15 

840 

26 

160 

811 

11 

3 

58 

665 

14 

865 

25 

135 

800 

11 

2 

59 

680 

15 

1 A 

891 

26 

109 

789 

11 

1 

60 

9.81 694 

14 

9.93 916 

25 

0.06 084 

9.87 778 

11 

0 


L Cos * 

d 

L Cot* 

c d 

L Tan 

L Sin* 

d 

/ 


Prop. Parts 


Subtract 10 from each en¬ 
try in the columns marked, 
with 



26 

25 

1 

2.6 

2.5 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 



15 

14 

1 

1.5 

1.4 

2 

3.0 

2.8 

3 

4.5 

4.2 

4 

6.0 

5.6 

5 

7.5 

7.0 

6 

9.0 

8.4 

7 

10.5 

9.8 

8 

12.0 

11.2 

9 

13.5 

12.6 



11 

10 

1 

1.1 

1.0 

2 

2.2 

2.0 

3 

3.3 

3.0 

4 

4.4 

4.0 

5 

5.5 

5.0 

6 

6.6 

6.0 

7 

7.7 

7.0 

8 

8.8 

8.0 

9 

9.9 

9.0 


[78] 


49' 





















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 41° 



48' 


[79] 




















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 42' 



[80] 


47 ' 








































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 43 



46' 


[81] 



















































IX. FIVE-PLACE LOGARITHMS OF FUNCTIONS — 44 c 


L Sin * d L Tan * c d L Cot 


L Cot 

L Cos* 

0.01 516 

9.85 693 

491 

681 

466 

669 

440 

657 

415 

645 

0.01 390 

9.85 632 

365 

620 

339 

608 

314 

596 

289 

583 

0.01 263 

9.85 571 

238 

559 

213 

547 

188 

534 

162 

522 

0.01 137 

9.85 510 

112 

497 

087 

485 

061 

473 

036 

460 

0.01 011 

9.85 448 

0.00 985 

436 

960 

423 

935 

411 

910 

399 

0.00 884 

9.85 386 

859 

374 

834 

361 

809 

349 

783 

337 

0.00 758 

9.85 324 

733 

312 

707 

299 

682 

287 

657 

274 

0.00 632 

9.85 262 

606 

250 

581 

237 

556 

225 

531 

212 

0.00 505 

9.85 200 

480 

187 

455 

175 

430 

162 

404 

150 

0.00 379 

9.85 137 

354 

125 

328 

112 

303 

100 

278 

087 

0.00 253 

9.85 074 

227 

062 

202 

049 

177 

037 

152 

024 

0.00 126 

9.85 012 

101 

9.84 999 

076 

986 

051 

974 

025 

961 

0.00 000 

9.84 949 

L Tan 

L Sin* 


Prop. Parts 


9.84 177 

190 
203 
216 
229 
9.84 242 

255 
269 
282 
295 
9.84 308 


321 
334 
347 
360 
9.84 373 

385 
398 
411 
424 
9.84 437 


450 
463 
476 
489 
9.84 502 

515 
528 
540 
553 
9.84 566 


579 
592 
605 
618 
9.84 630 

643 
656 
669 
682 
9.84 694 


707 
720 
733 
745 
9.84 758 

771 
784 
796 
809 
9.84 822 


835 
847 
860 
873 
9.84 885 

898 
911 
923 
936 
9.84 949 


L Cos* 


9.98 484 

509 
534 
560 
585 
9.98 610 

635 
661 
686 
711 
9.98 737 


762 
787 
812 
838 
9.98 863 

888 
913 
939 
964 
9.98 989 


9.99 015 
040 
065 
090 
9.99 116 

141 
166 
191 
217 
9.99 242 


267 
293 
318 
343 
9.99 368 

394 
419 
444 
469 
9.99 495 


520 
545 
570 
596 
'.99 621 

646 
672 
697 
722 
.99 747 


773 
798 
823 
848 
'.99 874 

899 
924 
949 
'.99 975 
i.OO 000 


L Cot* 


c d 


60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


Subtract 10 from each en¬ 
try in the columns marked 
with "*.” 



26 

25 

1 

2.6 

2.5 

2 

5.2 

5.0 

3 

7.8 

7.5 

4 

10.4 

10.0 

5 

13.0 

12.5 

6 

15.6 

15.0 

7 

18.2 

17.5 

8 

20.8 

20.0 

9 

23.4 

22.5 



14 

1 

1.4 

2 

2.8 

3 

4.2 

4 

5.6 

5 

7.0 

6 

8.4 

7 

9.8 

8 

11.2 

9 

12.6 



13 

12 

1 

1.3 

1.2 

2 

2.6 

2.4 

3 

3.9 

3.6 

4 

5.2 

4.8 

5 

6.5 

6.0 

6 

7.8 

7.2 

7 

9.1 

8.4 

8 

10.4 

9.6 

9 

11.7 

10.8 


45 c 


[82] 
















































X. AUXILIARY TABLES FOR ANGLES NEAR TO 0° OR 90° 

Note 1. On account of large tabular differences, results obtained by interpolation in 
the preceding table are particularly inaccurate in case we are determining log sin, log 
tan, or log cot for an angle near to 0°, or log cos, log tan, or log cot for an angle near to 
90°. To avoid such interpolation in Table IX, we use Table Xa or Table Xb. 

Table Xb applies without interpolation if the angle is expressed to the nearest tenth 
of a minute, and if the angle differs from 0°, or 90°, by at most 2°. If angles are ex¬ 
pressible to the nearest second, interpolation in Table Xb is sufficiently accurate except 
on the upper parts of pages 84 and 85. 

Table Xa applies for the indicated values of the angle M in connection with the 
following formulas.* 

A. Angle a near to 0°. Let M be the number of minutes in a. Then, 

log sin a = S + log M\ log tan a = T + log M. (1) 

B. Angle a near to 90°. Let M be the number of minutes in (90° — a). Then, 

log cos a = S + log M; log cot a = T + log M. (2) 

In using (1) and (2), we find S and T from Table Xa below and log M from 
Table VIII. 

Note 2. If log cot a is desired in (A), or log tan a in (B), recall that tan a = —— • 

cot a 


Table Xa. Auxiliary Table for S and T 


To find S or T, subtract 10 from the given entry. 


M 

S + 10 


M 

T + 10 

M 

7+10 

O' - 13' 

6.46373 


0' - 26' 

6.46373 

131' - 133' 

6.46394 

14' - 42' 

72 


27' - 39' 

74 

134' - 136' 

95 

43' - 58' 

71 


O 

1 

00 

75 

137' - 139' 

96 

59' - 71' 

6.46370 


49' - 56' 

6.46376 

140' - 142' 

6.46397 

72' - 81' 

69 


57' - 63' 

77 

143' - 145' 

98 

82' - 91' 

68 


64' - 69' 

78 

146' - 148' 

99 

92' - 99' 

6.46367 


o 

1 

4^ 

6.46379 

149'- 150' 

6.46400 

100' - 107' 

66 


75' - 80' 

80 

151' - 153' 

01 

108' - 115' 

65 


81' - 85' 

81 

154' - 156' 

02 

116' - 121' 

6.46364 


86' - 89' 

6.46382 

157' - 158' 

6.46403 

122' - 128' 

63 


90' - 94' 

83 

159' - 161' 

04 

129' - 134' 

62 


95' - 98' 

84 

162' - 163' 

05 

135' - 140' 

6.46361 


99' - 102' 

6.46385 

164' - 166' 

6.46406 

141' - 146' 

60 


103' - 106' 

86 

167' - 168' 

07 

147' - 151' 

59 


107' - 110' 

87 

169' - 171' 

08 

152' - 157' 

6.46358 


111' - 113' 

6.46388 

172' - 173' 

6.46409 

Or 

00 

1 

o> 

to 

57 


114' - 117' 

89 

174' - 175' 

10 

163' - 167' 

56 


118' - 120' 

90 

176' - 178' 

11 

168' - 171' 

6.46355 


121'- 124' 

6.46391 

179' - 180' 

6.46412 

172' - 176' 

54 


125' - 127' 

92 

181' - 182' 

13 

177' - 180' 

53 


128' - 130' 

93 

183' - 184' 

14 


* For an explanation of these formulas, see Hart’s Trigonometry, page 176. D. C. Heath and 
Company, publishers, 


[83] 

















Xb. ANGLES NEAR TO 0° OR 90° 


10 + log sin: 


/ 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

1.0 


n 

5. 

46373 

76476 

94085 

*06579 

*16270 

*24188 

*30882 

*36682 

*41797 

*46373 

59 

u 

1 

6. 46373 

50512 

54291 

57767 

60985 

63982 

66785 

69418 

71900 

74248 

76476 

58 

1 

o 

6. 76476 

78595 

80615 

82545 

84394 

86167 

87870 

89509 

91088 

92612 

94085 

57 

Q 

6. 94085 

95509 

96888 

98224 

99520 

*00779 

*02003 

*03193 

*04351 

*05479 

*06579 

56 

O 

4 

7. 06579 

07651 

08698 

09719 

10718 

11694 

12648 

13582 

14497 

15392 

16270 

55 

K 

7. 16270 

17130 

17973 

18800 

19612 

20409 

21191 

21960 

22715 

23458 

24188 

54 

A 

7. 24188 

24906 

25612 

26307 

26991 

27664 

28327 

28980 

29623 

30257 

30882 

53 

U 

7 

7. 30882 

31498 

32106 

32705 

33296 

33879 

34454 

35022 

35582 

36135 

36682 

52 

i 

Q 

7. 36682 

37221 

37754 

38280 

38800 

39314 

39822 

40324 

40821 

41312 

41797 

51 

9 

7. 41797 

42277 

42751 

43221 

43685 

44145 

44600 

45050 

45495 

45936 

46373 

50 

1 ft 

7. 46373 

46805 

47233 

47656 

48076 

48491 

48903 

49311 

49715 

50115 

50512 

49 

1 1 

7. 50512 

50905 

51294 

51680 

52063 

52442 

52818 

53191 

53561 

53927 

54291 

48 

1 9 

7. 54291 

54651 

55009 

55363 

55715 

56064 

56410 

56753 

57094 

57431 

57767 

47 


7. 57767 

58100 

58430 

58758 

59083 

59406 

59726 

60045 

60360 

60674 

60985 

46 

14 

7. 60985 

61294 

61601 

61906 

62209 

62509 

62808 

63104 

63399 

63691 

63982 

45 

15 

7. 63982 

64270 

64557 

64842 

65125 

65406 

65685 

65962 

66238 

66512 

66784 

44 

1 ft 

7. 66784 

67055 

67324 

67591 

67857 

68121 

68383 

68644 

68903 

69161 

69417 

43 

17 

7. 69417 

69672 

69925 

70177 

70427 

70676 

70924 

71170 

71414 

71658 

71900 

42 

13 

7. 71900 

72140 

72380 

72618 

72854 

73090 

73324 

73557 

73788 

74019 

74248 

41 

19 

7. 74248 

74476 

74703 

74928 

75153 

75376 

75598 

75819 

76039 

76258 

76475 

40 

20 

7.7 6475 

6692 

6907 

7122 

7335 

7548 

7759 

7969 

8179 

8387 

8594 

39 

21 

7.7 8594 

8801 

9006 

9210 

9414 

9616 

9818 

*0018 

*0218 

*0417 

*0615 

38 

22 

7.8 0615 

0812 

1008 

1203 

1397 

1591 

1783 

1975 

2166 

2356 

2545 

37 

23 

7.8 2545 

2733 

2921 

3108 

3294 

3479 

3663 

3847 

4030 

4212 

4393 

36 

24 

7.8 4393 

4574 

4754 

4933 

5111 

5289 

5466 

5642 

5817 

5992 

6166 

35 

25 

7.8 6166 

6340 

6512 

6684 

6856 

7026 

7196 

7366 

7534 

7702 

7870 

34 

26 

7.8 7870 

8036 

8202 

8368 

8533 

8697 

8860 

9023 

9186 

9347 

9509 

33 

27 

7.8 9509 

9669 

9829 

9988 

*0147 

*0305 

*0463 

*0620 

*0777 

*0933 

*1088 

32 

28 

7.9 1088 

1243 

1397 

1551 

1704 

1857 

2009 

2160 

2311 

2462 

2612 

31 

29 

7.9 2612 

2761 

2910 

3059 

3207 

3354 

3501 

3648 

3794 

3939 

4084 

30 

30 

7.9 4084 

4229 

4373 

4516 

4659 

4802 

4944 

5086 

5227 

5368 

5508 

29 

31 

7.9 5508 

5648 

5787 

5926 

6065 

6203 

6341 

6478 

6615 

6751 

6887 

28 

32 

7.9 6887 

7022 

7158 

7292 

7426 

7560 

7694 

7827 

7959 

8092 

8223 

27 

33 

7.9 8223 

8355 

8486 

8616 

8747 

8876 

9006 

9135 

9264 

9392 

9520 

26 

34 

7.9 9520 

9647 

9775 

9901 

*0028 

*0154 

*0279 

*0405 

*0530 

*0654 

*0779 

25 

35 

8.0 0779 

0903 

1026 

1149 

1272 

1395 

1517 

1639 

1760 

1881 

2002 

24 

36 

8.0 2002 

2123 

2243 

2362 

2482 

2601 

2720 

2838 

2957 

3074 

3192 

23 

37 

8.0 3192 

3309 

3426 

3543 

3659 

3775 

3891 

4006 

4121 

4236 

4350 

22 

38 

8.0 4350 

4464 

4578 

4692 

4805 

4918 

5030 

5143 

5255 

5367 

5478 

21 

39 

8.0 5478 

5589 

5700 

5811 

5921 

6031 

6141 

6251 

6360 

6469 

6578 

20 

40 

8.0 6578 

6686 

6794 

6902 

7010 

7117 

7224 

7331 

7438 

7544 

7650 

19 

41 

8.0 7650 

7756 

7861 

7967 

8072 

8176 

8281 

8385 

8489 

8593 

8696 

18 

42 

8.0 8696 

8800 

8903 

9006 

9108 

9210 

9312 

9414 

9516 

9617 

9718 

17 

43 

8.0 9718 

9819 

9920 

*0020 

*0120 

*0220 

*0320 

*0420 

*0519 

*0618 

*0717 

16 

44 

8.1 0717 

0815 

0914 

1012 

1110 

1207 

1305 

1402 

1499 

1596 

1693 

15 

45 

8.1 1693 

1789 

1885 

1981 

2077 

2172 

2268 

2363 

2458 

2553 

2647 

14 

46 

8.1 2647 

2741 

2836 

2929 

3023 

3117 

3210 

3303 

3396 

3489 

3581 

13 

47 

8.1 3581 

3673 

3765 

3857 

3949 

4041 

4132 

4223 

4314 

4405 

4495 

12 

48 

8.1 4495 

4586 

4676 

4766 

4856 

4945 

5035 

5124 

5213 

5302 

5391 

11 

49 

8.1 5391 

5479 

5568 

5656 

5744 

5832 

5919 

6007 

6094 

6181 

6268 

10 

50 

8.1 6268 

6355 

6441 

6528 

6614 

6700 

6786 

6872 

6957 

7043 

7128 

9 

51 

8.1 7128 

7213 

7298 

7383 

7467 

7552 

7636 

7720 

7804 

7888 

7971 

8 

52 

8.1 7971 

8055 

8138 

8221 

8304 

8387 

8469 

8552 

8634 

8716 

8798 

7 

53 

8.1 8798 

8880 

8962 

9044 

9125 

9206 

9287 

9368 

9449 

9530 

9610 

6 

54 

8.1 9610 

9691 

9771 

9851 

9931 

*0010 

*0090 

*0170 

*0249 

*0328 

*0407 

5 

55 

8.2 0407 

0486 

0565 

0643 

0722 

0800 

0878 

0956 

1034 

1112 

1189 

4 

56 

8.2 1189 

1267 

1344 

1422 

1499 

1576 

1652 

1729 

1805 

1882 

1958 

3 

57 

8.2 1958 

2034 

2110 

2186 

2262 

2337 

2413 

2488 

2563 

2638 

2713 

2 

58 

8.2 2713 

2788 

2863 

2937 

3012 

3086 

3160 

3234 

3308 

3382 

3456 

1 

59 

8.2 3456 

3529 

3603 

3676 

3749 

3822 

3895 

3968 

4041 

4113 

4186 

0 


1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

/ 


10 + log cos: 


89 ° 


[ 84 ] 





























































































Xb. ANGLES NEAR TO 0° OR 90 c 


19 





10 + log tan : 

0 C 




- j 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

1.0 

5 . 

6 . 46373 
6 . 76476 

6 . 94085 

7 . 06579 

7 . 16270 
7 . 24188 
7 . 30882 
7 . 36682 
7 . 41797 

46373 

50512 

78595 

95509 

07651 

17130 

24906 

31499 

37221 

42277 

76476 

54291 

80615 

96888 

08698 

17973 

25612 

32106 

37754 

42751 

94085 

57767 

82545 

98224 

09719 

18800 

26307 

32705 

38281 

43221 

*06579 

60985 

84394 

99521 

10718 

19612 

26991 

33296 

38801 

43686 

*16270 

63982 

86167 

*00779 

11694 

20409 

27664 

33879 

39315 

44145 

*24188 

66785 

87870 

*02003 

12648 

21191 

28327 

34454 

39823 

44600 

*30882 

69418 

89509 

*03193 

13582 

21960 

28980 

35022 

40325 

45050 

*36682 

71900 

91088 

*04351 

14497 

22715 

29624 

35582 

40821 

45495 

*41797 

74248 

92612 

*05479 

15392 

23458 

30258 

36135 

41312 

45936 

*46373 

76476 

94085 

*06579 

16270 

24188 

30882 

36682 

41797 

46373 

7 . 46373 
7 . 50512 
7 . 54291 
7 . 57767 
7 . 60986 

7 . 63982 
7 . 66785 
7 . 69418 
7 . 71900 
7 . 74248 

46805 

50905 

54651 

58100 

61295 

64271 

67056 

69673 

72141 

74476 

47233 

51295 

55009 

58430 

61602 

64557 

67324 

69926 

72380 

74703 

47656 

51681 

55363 

58758 

61906 

64842 

67592 

70178 

72618 

74929 

48076 

52063 

55715 

59083 

62209 

65125 

67857 

70428 

72855 

75153 

48492 

52443 

56064 

59406 

62510 

65406 

68121 

70677 

73090 

75377 

48903 

52819 

56410 

59727 

62808 

65685 

68384 

70924 

73324 

75599 

49311 

53191 

56753 

60045 

63105 

65963 

68645 

71170 

73557 

75820 

49715 

53561 

57094 

60361 

63399 

66239 

68904 

71415 

73789 

76040 

50115 

53927 

57432 

60674 

63692 

66513 

69162 

71658 

74019 

76258 

50512 

54291 

57767 

60986 

63982 

66785 

69418 

71900 

74248 

76476 

7.7 6476 

6693 

6908 

7123 

7336 

7549 

7760 

7970 

8179 

8388 

8595 

7.7 8595 

8801 

9007 

9211 

9415 

9617 

9819 

*0019 

*0219 

*0418 

*0615 1 

7.8 0615 

0812 

1009 

1204 

1398 

1591 

1784 

1976 

2167 

2357 

2546 

7.8 2546 

2734 

2922 

3109 

3295 

3480 

3664 

3848 

4031 

4213 

4394 

7.8 4394 

4575 

4755 

4934 

5112 

5290 

5467 

5643 

5819 

5993 

6167 I , 

7.8 6167 

6341 

6513 

6685 

6857 

7027 

7197 

7367 

7535 

7703 

7871 

7.8 7871 

8037 

8204 

8369 

8534 

8698 

8862 

9025 

9187 

9349 

9510 

7.8 9510 

9670 

9830 

9990 

*0149 

*0307 

*0464 

*0622 

*0778 

*0934 

*1089 

7.9 1089 

1244 

1398 

1552 

1705 

1858 

2010 

2162 

2313 

2463 

2613 

7.9 2613 

2763 

2912 

3060 

3208 

3356 

3503 

3649 

3795 

3941 

4086 

7.9 4086 

4230 

4374 

4518 

4661 

4804 

4946 

5088 

5229 

5370 

5510 ^ 

7.9 5510 

5650 

5789 

5928 

6067 

6205 

6343 

6480 

6617 

6753 

6889 1 

7.9 6889 

7024 

7159 

7294 

7428 

7562 

7696 

7829 

7961 

8094 

8225 S 

7.9 8225 

8357 

8488 

8618 

8749 

8878 

9008 

9137 

9266 

9394 

9522 S 

7.9 9522 

9649 

9777 

9903 

*0030 

*0156 

*0282 

*0407 

*0532 

*0657 

*0781 J 

8.0 0781 

0905 

1028 

1152 

1274 

1397 

1519 

1641 

1762 

1884 

2004 S 

8.0 2004 

2125 

2245 

2365 

2484 

2604 

2722 

2841 

2959 

3077 

3194 £ 

8.0 3194 

3312 

3429 

3545 

3661 

3777 

3893 

4008 

4124 

4238 

4353 2 

8.0 4355 

4467 

4581 

4694 

4808 

4921 

5033 

5146 

5258 

5369 

5481 1 2 

8.0 5481 

5592 

5703 

5814 

5924 

6034 

6144 

6254 

6363 

6472 

6581 2 

8.0 6581 

6689 

6797 

6905 

7013 

7120 

7227 

7334 

7441 

7547 

7653 1 

8.0 7653 

7759 

7864 

7970 

8075 

8180 

8284 

8388 

8492 

8596 

8700 1 

8.0 8700 

8803 

8906 

9009 

9111 

9214 

9316 

9418 

9519 

9621 

9722 1 1 

8.0 9722 

9823 

9923 

*0024 

*0124 

*0224 

*0324 

*0423 

*0522 

*0621 

*0720 1 

8.1 0720 

0819 

0917 

1015 

1113 

1211 

1309 

1406 

1503 

1600 

1696 1 

8.1 1696 

1793 

1889 

1985 

2081 

2176 

2272 

2367 

2462 

2556 

2651 1 

8.1 2651 

2745 

2839 

2933 

3027 

3121 

3214 

3307 

3400 

3493 

3585 1 

8.1 3585 

3677 

3770 

3861 

3953 

4045 

4136 

4227 

4318 

4409 

4500 1 

8.1 4500 

4590 

4680 

4770 

4860 

4950 

5039 

5128 

5218 

5306 

5395 1 

8.1 5395 

5484 

5572 

5660 

5748 

5836 

5924 

6011 

6099 

6186 

6273 1 

8.1 6273 

6359 

6446 

6533 

6619 

6705 

6791 

6877 

6962 

7048 

7133 

8.1 7133 

7218 

7303 

7388 

7472 

7557 

7641 

7725 

7809 

7893 

7976 

8.1 7976 

8060 

8143 

8226 

8309 

8392 

8475 

8557 

8639 

8722 

8804 

8.1 8804 

8886 

8967 

9049 

9130 

9211 

9293 

9374 

9454 

9535 

9616 

8.1 9616 

9696 

9776 

9856 

9936 

*0016 

*0096 

*0175 

*0254 

*0334 

*0413 

8.2 0413 

0491 

0570 

0649 

0727 

0806 

0884 

0962 

1040 

1118 

1195 

8.2 1195 

1273 

1350 

1427 

1504 

1581 

1658 

1735 

1811 

1888 

1964 

8.2 1964 

2040 

2116 

2192 

2268 

2343 

2419 

2494 

2569 

2645 

2720 

8.2 2720 

2794 

2869 

2944 

3018 

3092 

3167 

3241 

3315 

3388 

3462 

8.2 3462 

3536 

3609 

3682 

3756 

3829 

3902 

3974 

4047 

4120 

4192 

1.0 

e9 

.8 

.7 

.6 

.5 

• 4 

.3 

.2 

.1 

.0 j ' 




10 + log cot : 

89 ° 




1 + 


[ 85 ] 































































































































Xb. ANGLES NEAR TO 0° OR 90° 


t 





10 + log sin: 

i ° 






/ 

.0 

.1 

.2 

.3 

.4 

•5 

.6 

.7 

.8 

.9 

1.0 


0 

8.2 

4186 

4258 

4330 

4402 

4474 

4546 

4618 

4689 

4761 

4832 

4903 

59 

1 

8.2 

4903 

4974 

5045 

5116 

5187 

5258 

5328 

5399 

5469 

5539 

5609 

58 

2 

8.2 

5609 

5679 

5749 

5819 

5889 

5958 

6028 

6097 

6166 

6235 

6304 

57 

3 

8.2 

6304 

6373 

6442 

6511 

6579 

6648 

6716 

6784 

6852 

6920 

6988 

56 

4 

8.2 

6988 

7056 

7124 

7191 

7259 

7326 

7393 

7460 

7528 

7595 

7661 

55 

5 

8.2 

7661 

7728 

7795 

7861 

7928 

7994 

8060 

8127 

8193 

8258 

8324 

54 

6- 

8.2 

8324 

8390 

8456 

8521 

8587 

8652 

8717 

8782 

8848 

8912 

8977 

53 

7 

8.2 

8977 

9042 

9107 

9171 

9236 

9300 

9364 

9429 

9493 

9557 

9621 

52 

8 

8.2 

9621 

9684 

9748 

9812 

9875 

9939 

*0002 

*0065 

*0129 

*0192 

*0255 

51 

9 

8.3 

0255 

0317 

0380 

0443 

0506 

0568 

0631 

0693 

0755 

0817 

0879 

50 

10 

8.3 

0879 

0941 

1003 

1065 

1127 

1188 

1250 

1311 

1373 

1434 

1495 

49 

11 

8.3 

1495 

1556 

1618 

1678 

1739 

1800 

1861 

1921 

1982 

2042 

2103 

48 

12 

8.3 

2103 

2163 

2223 

2283 

2343 

2403 

2463 

2523 

2583 

2642 

2702 

47 

13 

8.3 

2702 

2761 

2820 

2880 

2939 

2998 

3057 

3116 

3175 

3234 

3292 

46 

14 

8.3 

3292 

3351 

3410 

3468 

3527 

3585 

3643 

3701 

3759 

3817 

3875 

45 

15 

8.3 

3875 

3933 

3991 

4049 

4106 

4164 

4221 

4279 

4336 

4393 

4450 

44 

16 

8.3 

4450 

4508 

4565 

4621 

4678 

4735 

4792 

4849 

4905 

4962 

5018 

43 

17 

8.3 

5018 

5074 

5131 

5187 

5243 

5299 

5355 

5411 

5467 

5523 

5578 

42 

18 

8.3 

5578 

5634 

5690 

5745 

5800 

5856 

5911 

5966 

6021 

6076 

6131 

41 

19 

8.3 

6131 

6186 

6241 

6296 

6351 

6405 

6460 

6515 

6569 

6623 

6678 

40 

20 

8.3 

6678 

6732 

6786 

6840 

6894 

6948 

7002 

7056 

7110 

7163 

7217 

39 

21 

8.3 

7217 

7271 

7324 

7378 

7431 

7484 

7538 

7591 

7644 

7697 

7750 

38 

22 

8.3 

7750 

7803 

7856 

7908 

7961 

8014 

8066 

8119 

8171 

8224 

8276 

37 

23 

8.3 

8276 

8328 

8381 

8433 

8485 

8537 

8589 

8641 

8693 

8744 

8796 

36 

24 

8.3 

8796 

8848 

8899 

8951 

9002 

9054 

9105 

9157 

9208 

9259 

9310 

35 

25 

8.3 

9310 

9361 

9412 

9463 

9514 

9565 

9616 

9666 

9717 

9767 

9818 

34 

26 

8.3 

9818 

9868 

9919 

9969 

*0019- 

*0070 

*0120 

*0170 

*0220 

*0270 

*0320 

33 

27 

8.4 

0320 

0370 

0420 

0469 

0519 

0569 

0618 

0668 

0717 

0767 

0816 

32 

28 

8.4 

0816 

0865 

0915 

0964 

1013 

1062 

mi 

1160 

1209 

1258 

1307 

31 

29 

8.4 

1307 

1356 

1404 

1453 

1501 

1550 

1598 

1647 

1695 

1744 

1792 

30 

30 

8.4 

1792 

1840 

1888 

1936 

1984 

2032 

2080 

2128 

2176 

2224 

2272 

29 

31 

8.4 

2272 

2319 

2367 

2415 

2462 

2510 

2557 

2604 

2652 

2699 

2746 

28 

32 

8.4 

2746 

2793 

2840 

2888 

2935 

2982 

3028 

3075 

3122 

3169 

3216 

27 

33 

8.4 

3216 

3262 

3309 

3355 

3402 

3448 

3495 

3541 

3588 

3634 

3680 

26 

34 

8.4 

3680 

3726 

3772 

3818 

3864 

3910 

3956 

4002 

4048 

4094 

4139 

25 

35 

8.4 

4139 

4185 

4231 

4276 

4322 

4367 

4413 

4458 

4504 

4549 

4594 

24 

36 

8.4 

4594 

4639 

4684 

4730 

4775 

4820 

4865 

4910 

4954 

4999 

5044 

23 

37 

8.4 

5044 

5089 

5133 

5178 

5223 

5267 

5312 

5356 

5401 

5445 

5489 

22 

38 

8.4 

5489 

5534 

5578 

5622 

5666 

5710 

5754 

5798 

5842 

5886 

5930 

21 

39 

8.4 

5930 

5974 

6018 

6061 

6105 

6149 

6192 

6236 

6280 

6323 

6366 

20 

40 

8.4 

6366 

6410 

6453 

6497 

6540 

6583 

6626 

6669 

6712 

6755 

6799 

19 

41 

8.4 

6799 

6841 

6884 

6927 

6970 

7013 

7056 

7098 

7141 

7184 

7226 

18 

42 

8.4 

7226 

7269 

7311 

7354 

7396 

7439 

7481 

7523 

7565 

7608 

7650 

17 

43 

8.4 

7650 

7692 

7734 

7776 

7818 

7860 

7902 

7944 

7986 

8028 

8069 

16 

44 

8.4 

8069 

8111 

8153 

8194 

8236 

8278 

8319 

8361 

8402 

8443 

8485 

15 

45 

8.4 

8485 

8526 

8567 

8609 

8650 

8691 

8732 

8773 

8814 

8855 

8896 

14 

46 

8.4 

8896 

8937 

8978 

9019 

9060 

9101 

9141 

9182 

9223 

9263 

9304 

13 

47 

8.4 

9304 

9345 

9385 

9426 

9466 

9506 

9547 

9587 

9627 

9668 

9708 

12 

48 

8.4 

9708 

9748 

9788 

9828 

9868 

9908 

9948 

9988 

*0028 

*0068 

*0108 

11 

49 

8.5 

0108 

0148 

0188 

0227 

0267 

0307 

0346 

0386 

0425 

0465 

0504 

10 

50 

8.5 

0504 

0544 

0583 

0623 

0662 

0701 

0741 

0780 

0819 

0858 

0897 

9 

51 

8.5 

0897 

0936 

0976 

1015 

1054 

1092 

1131 

1170 

1209 

1248 

1287 

8 

52 

8.5 

1287 

1325 

1364 

1403 

1442 

1480 

1519 

1557 

1596 

1634 

1673 

7 

53 

8.5 

1673 

1711 

1749 

1788 

1826 

1864 

1903 

1941 

1979 

2017 

2055 

6 

54 

8.5 

2055 

2093 

2131 

2169 

2207 

2245 

2283 

2321 

2359 

2397 

2434 

5 

55 

8.5 

2434 

2472 

2510 

2547 

2585 

2623 

2660 

2698 

2735 

2773 

2810 

4 

56 

8.5 

2810 

2848 

2885 

2922 

2960 

2997 

3034 

3071 

3109 

3146 

3183 

3 

57 

8.5 

3183 

3220 

3257 

3294 

3331 

3368 

3405 

3442 

3479 

3515 

3552 

2 

58 

8.5 

3552 

3589 

3626 

3663 

3699 

3736 

3772 

3809 

3846 

3882 

3919 

1 

59 

8.5 

3919 

3955 

3992 

4028 

4064 

4101 

4137 

4173 

4210 

4246 

4282 

0 


1.0 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

/ j 






10 + log cos: 

88 ° 





* 


[ 86 ] 























































































Xb. ANGLES NEAR TO 0° OR 90° 


21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 





10 + log tan: 

r 

D 




.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

1.0 

) 8.2 4192 

8.2 4910 

2 8.2 5616 

2 8.2 6312 

t 8.2 6996 

8.2 7669 
) 8.2 8332 

8.2 8986 

8.2 9629 

8.3 0263 

4264 

4981 

5686 

6380 

7063 

7736 

8398 

9050 

9693 

0326 

4337 

5052 

5756 

6449 

7131 

7803 

8464 

9115 

9757 

0389 

4409 

5123 

5826 

6518 

7199 

7869 

8529 

9180 

9820 

0452 

4481 

5194 

5896 

6586 

7266 

7936 

8595 

9244 

9884 

0514 

4553 

5265 

5965 

6655 

7334 

8002 

8660 

9309 

9947 

0577 

4624 

5335 

6035 

6723 

7401 

8068 

8725 

9373 

*0011 

0639 

4696 

5406 

6104 

6792 

7468 

8134 

8791 

9437 

*0074 

0702 

4767 

5476 

6173 

6860 

7535 

8201 

8856 

9501 

*0137 

0764 

4839 

5546 

6243 

6928 

7602 

8266 

8921 

9565 

*0200 

0826 

4910 

5616 

6312 

6996 

7669 

8332 

8986 

9629 

*0263 

0888 

8.3 0888 
8.3 1505 
8.3 2112 
8.3 2711 
8.3 3302 

8.3 3886 
8.3 4461 
8.3 5029 
8.3 5590 
8.3 6143 

0950 

1566 

2173 

2771 

3361 

3944 

4518 

5085 

5645 

6198 

1012 

1627 

2233 

2830 

3420 

4001 

4575 

5142 

5701 

6253 

1074 

1688 

2293 

2890 

3478 

4059 

4632 

5198 

5756 

6308 

1136 

1749 

2353 

2949 

3537 

4117 

4689 

5254 

5812 

6362 

1198 

1809 

2413 

3008 

3595 

4174 

4746 

5310 

5867 

6417 

1259 

1870 

2473 

3067 

3653 

4232 

4803 

5366 

5922 

6472 

1321 

1931 

2533 

3126 

3712 

4289 

4859 

5422 

5978 

6526 

1382 

1991 

2592 

3185 

3770 

4347 

4916 

5478 

6033 

6581 

1443 

2052 

2652 

3244 

3828 

4404 

4972 

5534 

6088 

6635 

1505 

2112 

2711 

3302 

3886 

4461 

5029 

5590 

6143 

6689 

8.3 6689 

6744 

6798 

6852 

6906 

6960 

7014 

7068 

7122 

7175 

7229 

8.3 7229 

7283 

7336 

7390 

7443 

7497 

7550 

7603 

7656 

7709 

7762 

8.3 7762 

7815 

7868 

7921 

7974 

8026 

8079 

8132 

8184 

8236 

8289 

8.3 8289 

8341 

8393 

8446 

8498 

8550 

8602 

8654 

8706 

8757 

8809 

8.3 8809 

8861 

8913 

8964 

9016 

9067 

9118 

9170 

9221 

9272 

9323 

8.3 9323 

9374 

9425 

9476 

9527 

9578 

9629 

9680 

9730 

9781 

9832 

8.3 9832 

9882 

9932 

9983 

*0033 

*0083 

*0134 

*0184 

*0234 

*0284 

*0334 

8.4 0334 

0384 

0434 

0483 

0533 

0583 

0632 

0682 

0732 

0781 

0830 

8.4 0830 

0880 

0929 

0978 

1027 

1077 

1126 

1175 

1224 

1272 

1321 

8.4 1321 

1370 

1419 

1468 

1516 

1565 

1613 

1662 

1710 

1758 

1807 

8.4 1807 

1855 

1903 

1951 

1999 

2048 

2095 

2143 

2191 

2239 

2287 

8.4 2287 

2335 

2382 

2430 

2477 

2525 

2572 

2620 

2667 

2715 

2762 

8.4 2762 

2809 

2856 

2903 

2950 

2997 

3044 

3091 

3138 

3185 

3232 

8.4 3232 

3278 

3325 

3371 

3418 

3464 

3511 

3557 

3604 

3650 

3696 

8.4 3696 

3742 

3789 

3835 

3881 

3927 

3973 

4019 

4064 

4110 

4156 

8.4 4156 

4202 

4247 

4293 

4339 

4384 

4430 

4475 

4520 

4566 

4611 

8.4 4611 

4656 

4701 

4747 

4792 

4837 

4882 

4927 

4972 

5016 

5061 

8.4 5061 

5106 

5151 

5195 

5240 

5285 

5329 

5374 

5418 

5463 

5507 

8.4 5507 

5551 

5596 

5640 

5684 

5728 

5772 

5816 

5860 

5904 

5948 

8.4 5948 

5992 

6036 

6080 

6123 

6167 

6211 

6254 

6298 

6341 

6385 

8.4 6385 

6428 

6472 

6515 

6558 

6602 

6645 

6688 

6731 

6774 

6817 

8.4 6817 

6860 

6903 

6946 

6989 

7032 

7075 

7117 

7160 

7203 

7245 

8.4 7245 

7288 

7330 

7373 

7415 

7458 

7500 

7543 

7585 

7627 

7669 

8.4 7669 

7712 

7754 

7796 

7838 

7880 

7922 

7964 

8006 

8047 

8089 

8.4 8089 

8131 

8173 

8214 

8256 

8298 

8339 

8381 

8422 

8464 

8505 

8.4 8505 

8546 

8588 

8629 

8670 

8711 

8753 

8794 

8835 

8876 

8917 

8.4 8917 

8958 

8999 

9040 

9081 

9121 

9162 

9203 

9244 

9284 

9325 

8.4 9325 

9366 

9406 

9447 

9487 

9528 

9568 

9608 

9649 

9689 

9729 

8.4 9729 

9769 

9810 

9850 

9890 

9930 

9970 

*0010 

*0050 

*0090 

*0130 

8.5 0130 

0170 

0209 

0249 

0289 

0329 

0368 

0408 

0448 

0487 

0527 3 

8.5 0527 

0566 

0606 

0645 

0684 

0724 

0763 

0802 

0842 

0881 

0920 

8.5 0920 

0959 

0998 

1037 

1076 

1115 

1154 

1193 

1232 

1271 

1310 

8.5 1310 

1349 

1387 

1426 

1465 

1503 

1542 

1581 

1619 

1658 

1696 

8.5 1696 

1735 

1773 

1811 

1850 

1888 

1926 

1964 

2003 

2041 

2079 

8.5 2079 

2117 

2155 

2193 

2231 

2269 

2307 

2345 

2383 

2421 

2459 

8.5 2459 

2496 

2534 

2572 

2610 

2647 

2685 

2722 

2760 

2797 

2835 

8.5 2835 

2872 

2910 

2947 

2985 

3022 

3059 

3096 

3134 

3171 

3208 

8.5 3208 

3245 

3282 

3319 

3356 

3393 

3430 

3467 

3504 

3541 

3578 

8.5 3578 

3615 

3651 

3688 

3725 

3762 

3798 

3835 

3872 

3908 

3945 

8.5 3945 

3981 

4018 

4054 

4091 

4127 

4163 

4200 

4236 

4272 

4308 

1.0 

.9 

.8 

.7 | 

.6 

.5 

.4 

.3 

.2 

.1 

.0 > 




10 + log cot 


88 ° 




\ 


59 

58 

57 

56 

55 

54 

53 

52 

51 

50 


48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 


9 

8 

7 

6 

5 

4 

3 

2 

1 

0 


[ 87 ] 




































































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 


->■ 

C 

1° 

3 

o 

2° 

3° 

4° 


/ 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.00000 

1.0000 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

60 

1 

029 

000 

774 

984 

519 

938 

263 

861 

.07005 

754 

59 

2 

058 

000 

803 

984 

548 

937 

292 

860 

034 

752 

£8 

3 

087 

000 

832 

983 

577 

936 

321 

858 

063 

750 

57 

4 

116 

000 

862 

983 

606 

935 

350 

857 

092 

748 

56 

5 

.00145 

1.0000 

.01891 

.99982 

.03635 

.99934 

.05379 

.99855 

.07121 

.99746 

55 

6 

175 

000 

920 

982 

664 

933 

408 

854 

150 

744 

54 

7 

204 

000 

949 

981 

693 

932 

437 

852 

179 

742 

53 

8 

233 

000 

.01978 

980 

723 

931 

466 

851 

208 

740 

52 

9 

262 

000 

.02007 

980 

752 

930 

495 

849 

237 

738 

51 

10 

.00291 

1.0000 

.02036 

.99979 

.03781 

.99929 

.05524 

.99847 

.07266 

.99736 

50 

11 

320 

.99999 

065 

979 

810 

927 

553 

846 

295 

734 

49 

12 

349 

999 

094 

978 

839 

926 

582 

844 

324 

731 

48 

13 

378 

999 

123 

977 

868 

925 

611 

842 

353 

729 

47 

14 

407 

999 

152 

977 

897 

924 

640 

841 

382 

727 

46 

15 

.00436 

.99999 

.02181 

.99976 

.03926 

.99923 

.05669 

.99839 

.07411 

.99725 

45 

16 

465 

999 

211 

976 

955 

922 

698 

838 

440 

723 

44 

17 

495 

999 

240 

975 

.03984 

921 

727 

836 

469 

721 

43 

18 

524 

999 

269 

974 

.04013 

919 

756 

834 

498 

719 

42 

19 

553 

998 

298 

974 

042 

918 

785 

833 

527 

716 

41 

20 

.00582 

.99998 

.02327 

.99973 

.04071 

.99917 

.05814 

.99831 

.07556 

.99714 

40 

21 

611 

998 

356 

972 

100 

916 

844 

829 

585 

712 

39 

22 

640 

998 

385 

972 

129 

915 

873 

827 

614 

710 

38 

23 

669 

998 

414 

971 

159 

913 

902 

826 

643 

708 

37 

24 

698 

998 

443 

970 

188 

912 

931 

824 

672 

705 

36 

25 

.00727 

.99997 

.02472 

.99969 

.04217 

.99911 

.05960 

.99822 

.07701 

.99703 

35 

26 

756 

997 

501 

969 

246 

910 

.05989 

821 

730 

701 

34 

27 

785 

997 

530 

968 

275 

909 

.06018 

819 

759 

699 

33 

28 

814 

997 

560 

967 

304 

907 

047 

817 

788 

696 

32 

29 

844 

996 

589 

966 

333 

906 

076 

815 

817 

694 

31 

30 

.00873 

.99996 

.02618 

.99966 

.04362 

.99905 

.06105 

.99813 

.07846 

.99692 

30 

31 

902 

996 

647 

965 

391 

904 

134 

812 

875 

689 

29 

32 

931 

996 

676 

964 

420 

902 

163 

810 

904 

687 

28 

33 

960 

995 

705 

963 

449 

901 

192 

808 

933 

685 

27 

34 

.00989 

995 

734 

963 

478 

900 

221 

806 

962 

683 

26 

35 

.01018 

.99995 

.02763 

.99962 

.04507 

.99898 

.06250 

.99804 

.07991 

.99680 

25 

36 

047 

995 

792 

961 

536 

897 

279 

803 

.08020 

678 

24 

37 

076 

994 

821 

960 

565 

896 

308 

801 

049 

676 

23 

38 

105 

994 

850 

959 

594 

894 

337 

799 

078 

673 

22 

39 

134 

994 

879 

959 

623 

893 

366 

797 

107 

671 

21 

40 

.01164 

.99993 

.02908 

.99958 

.04653 

.99892 

.06395 

.99795 

.08136 

.99668 

20 

41 

193 

993 

938 

957 

682 

890 

424 

793 

165 

666 

19 

42 

222 

993 

967 

956 

711 

889 

453 

792 

194 

664 

18 

43 

251 

992 

.02996 

955 

740 

888 

482 

790 

223 

661 

17 

44 

280 

992 

.03025 

954 

769 

886 

511 

788 

252 

659 

16 

45 

.01309 

.99991 

.03054 

.99953 

.04798 

.99885 

.06540 

.99786 

.08281 

.99657 

15 

46 

338 

991 

083 

952 

827 

883 

569 

784 

310 

654 

14 

47 

367 

991 

112 

952 

856 

882 

598 

782 

339 

652 

13 

48 

396 

990 

141 

951 

885 

881 

627 

780 

368 

649 

12 

49 

425 

990 

170 

950 

914 

879 

656 

778 

397 

647 

11 

50 

.01454 

.99989 

.03199 

.99949 

.04943 

.99878 

.06685 

.99776 

.08426 

.99644 

10 

51 

483 

989 

228 

948 

.04972 

876 

714 

774 

455 

642 

9 

52 

513 

989 

257 

947 

.05001 

875 

743 

772 

484 

639 

8 

53 

542 

988 

286 

946 

030 

873 

773 

770 

513 

637 

7 

54 

571 

988 

316 

945 

059 

872 

802 

768 

542 

635 

6 

55 

.01600 

.99987 

.03345 

.99944 

.05088 

.99870 

.06831 

.99766 

.08571 

.99632 

5 

56 

629 

987 

374 

943 

117 

869 

~ 860 

764 

600 

630 

4 

57 

658 

986 

403 

942 

146 

867 

889 

762 

629 

627 

3 

58 

687 

986 

432 

941 

175 

866 

918 

760 

658 

625 

2 

59 

716 

985 

461 

940 

205 

864 

947 

758 

687 

622 

1 

60 

.01745 

.99985 

.03490 

.99939 

.05234 

.99863 

.06976 

.99756 

.08716 

.99619 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


89° 

o 

CO 

CO 

o 

C- 

CO 

86 

0 

85 

o 



[ 88 ] 
























































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



0° 

1° 

2° 

3° 

4° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

1 

2 

3 

4 

5 

6 

7 

8 

1 9 
10 

.00000 

029 

058 

087 

116 

.00145 

175 

204 

233 

262 

.00291 

3437.7 

1718.9 

1145.9 
859.44 
687.55 

572.96 
491.11 
429.72 

381.97 
343.77 

.01746 

775 

804 

833 

862 

.01891 

920 

949 

.01978 

.02007 

.02036 

57.290 

56.351 

55.442 

54.561 

53.709 

52.882 

52.081 

51.303 

50.549 

49.816 

49.104 

.03492 

521 

550 

579 

609 

.03638 

667 

696 

725 

754 

.03783 

28.636 

.399 

28.166 

27.937 

.712 

27.490 

.271 

27.057 

26.845 

.637 

26.432 

.05241 

270 

299 

328 

357 

.05387 

416 

445 

474 

503 

.05533 

19.081 

18.976 

. 871 . 

.768 

.666 

18.564 

.464 

.366 

.268 

.171 

18.075 

.06993 

.07022 

051 

080 

110 

.07139 

168 

197 

227 

256 

.07285 

14.301 

.241 

.182 

.124 

.065 

14.008 

13.951 

.894 

.838 

.782 

13.727 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

11 

320 

312.52 

066 

48.412 

812 

.230 

562 

17.980 

314 

672 

49 

12 

349 

286.48 

095 

47.740 

842 

26.031 

591 

.886 

344 

617 

48 

13 

378 

264.44 

124 

47.085 

871 

25.835 

620 

.793 

373 

563 

47 

14 

4 u / 

245.55 

153 

46.449 

900 

.642 

649 

• .702 

402 

510 

46 

15 

.00436 

229.18 

.02182 

45.829 

.03929 

25.452 

.05678 

17.611 

.07431 

13.457 

45 

16 

465 

214.86 

211 

45.226 

958 

.264 

708 

.521 

461 

.404 

44 

17 

495 

202.22 

240 

44.639 

.03987 

25.080 

737 

.431 

490 

.352 

43 

18 

524 

190.98 

269 

44.066 

.04016 

24.898 

766 

.343 

519 

.300 

42 

19 

553 

180.93 

298 

43.508 

046 

.719 

795 

.256 

548 

.248 

41 

20 

.00582 

171.89 

.02328 

42.964 

.04075 

24.542 

.05824 

17.169 

.07578 

13.197 

40 

21 

611 

163.70 

357 

42.433 

104 

.368 

854 

17.084 

607 

.146 

39 

22 

640 

156.26 

386 

41.916 

133 

.196 

883 

16.999 

636 

.096 

38 

23 

669 

149.47 

415 

41.411 

162 

24.026 

912 

.915 

665 

13.046 

37 

24 

698 

143.24 

444 

40.917 

191 

23.859 

941 

.832 

695 

12.996 

36 

25 

.00727 

137.51 

.02473 

40.436 

.04220 

23.695 

.05970 

16.750 

.07724 

12.947 

35 

26 

756 

132.22 

502 

39.965 

250 

.532 

.05999 

.668 

753 

.898 

34 

27 

785 

127.32 

531 

39.506 

279 

.372 

.06029 

.587 

782 

.850 

33 

28 

815 

122.77 

560 

39.057 

308 

.214 

058 

.507 

812 

.801 

32 

29 

844 

118.54 

589 

38.618 

337 

23.058 

087 

.428 

841 

.754 

31 

30 

.00873 

114.59 

.02619 

38.188 

.04366 

22.904 

.06116 

16.350 

.07870 

12.706 

30 

31 

902 

110 . 89 . 

648 

37.769 

395 

.752 

145 

.272 

899 

.659 

29 

32 

931 

107.43 

677 

37.358 

424 

.602 

175 

.195 

929 

.612 

28 

33 

960 

104.17 

706 

36.956 

454 

.454 

204 

.119 

958 

.566 

27 

34 

.00989 

101.11 

735 

36.563 

483 

.308 

233 

16.043 

.07987 

.520 

26 

35 

.01018 

98.218 

.02764 

36.178 

.04512 

22.164 

.06262 

15.969 

.08017 

12.474 

25 

36 

047 

95.489 

793 

35.801 

541 

22.022 

291 

.895 

046 

.429 

24 

37 

076 

92.908 

822 

35.431 

570 

21.881 

321 

.821 

075 

.384 

23 

38 

105 

90.463 

851 

35.070 

599 

.743 

350 

.748 

104 

.339 

22 

39 

135 

88.144 

881 

34.715 

628 

.606 

379 

.676 

134 

.295 

21 

40 

.01164 

85.940 

.02910 

34.368 

.04658 

21.470 

.06408 

15.605 

.08163 

12.251 

20 

41 

193 

83.844 

939 

34.027 

687 

.337 

438 

.534 

192 

.207 

19 

42 

222 

81.847 

968 

33.694 

716 

.205 

467 

.464 

221 

.163 

18 

43 

251 

79.943 

.02997 

33.366 

745 

21.075 

496 

.394 

251 

.120 

17 

44 

280 

78.126 

.03026 

33.045 

774 

20.946 

525 

.325 

280 

.077 

16 

45 

.01309 

76.390 

.03055 

32.730 

.04803 

20.819 

.06554 

15.257 

.08309 

12.035 

15 

46 

338 

74.729 

084 

32.421 

833 

.693 

584 

.189 

339 

11.992 

14 

47 

367 

73.139 

114 

32.118 

862 

.569 

613 

.122 

368 

.950 

13 

48 

396 

71.615 

143 

31.821 

891 

.446 

642 

15.056 

397 

.909 

12 

49 

425 

70.153 

172 

31.528 

920 

.325 

671 

14.990 

427 

.867 

11 

50 

.01455 

68.750 

.03201 

31.242 

.04949 

20.206 

.06700 

14.924 

.08456 

11.826 

10 

51 

484 

67.402 

230 

30.960 

.04978 

20.087 

730 

.860 

485 

.785 

9 

52 

513 

66.105 

259 

30.683 

.05007 

19.970 

759 

.795 

514 

.745 

8 

53 

542 

64.858 

288 

30.412 

037 

.855 

788 

.732 

544 

.705 

7 

54 

571 

63.657 

317 

30.145 

066 

.740 

817 

.669 

573 

.664 

6 

55 

.01600 

62.499 

.03346 

29.882 

.05095 

19.627 

.06847 

14.606 

.08602 

11.625 

5 

56 

629 

61.383 

376 

29.624 

124 

.516 

876 

.544 

632 

.585 

4 

57 

658 

60.306 

405 

29.371 

153 

.405 

905 

.482 

661 

.546 

3 

58 

687 

59.266 

434 

29.122 

182 

.296 

934 

.421 

690 

.507 

2 

59 

716 

58.261 

463 

28.877 

212 

.188 

963 

.361 

720 

.468 

1 

60 

.01746 

57.290 

.03492 

28.636 

.05241 

19.081 

.06993 

14.301 

.08749 

11.430 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 


89 


88 

o 

87 

0 

86 

D 

85 

0 



[ 89 ] 





















































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



5 

° 

6 

° 

7 

ro 

8° 

9° 


/ 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.08716 

.99619 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

60 

1 

745 

617 

482 

449 

216 

251 

946 

023 

672 

764 

59 

2 

774 

614 

511 

446 

245 

248 

.13975 

019 

701 

760 

58 

3 

803 

612 

540 

443 

274 

244 

.14004 

015 

730 

755 

57 

4 

831 

609 

569 

440 

302 

240 

033 

011 

758 

751 

56 

5 

.08860 

.99607 

.10597 

.99437 

.12331 

.99237 

.14061 

.99006 

.15787 

.98746 

55 

6 

889 

604 

626 

434 

360 

233 

090 

.99002 

816 

741 

54 

7 

918 

602 

655 

431 

389 

230 

119 

.98998 

845 

737 

53 

8 

947 

599 

684 

428 

418 

226 

148 

994 

873 

732 

52 

9 

.08976 

596 

713 

424 

447 

222 

177 

990 

902 

728 

51 

10 

.09005 

.99594 

.10742 

.99421 

.12476 

.99219 

.14205 

.98986 

.15931 

.98723 

50 

11 

034 

591 

771 

418 

504 

215 

234 

982 

959 

718 

49 

12 

063 

588 

800 

415 

533 

211 

263 

978 

.15988 

714 

48 

13 

092 

586 

829 

412 

562 

208 

.292 

973 

.16017 

709 

47 

14 

121 

583 

858 

409 

591 

204 

320 

969 

046 

704 

46 

15 

.09150 

.99580 

.10887 

.99406 

.12620 

.99200 

.14349 

.98965 

.16074 

.98700 

45 

16 

179 

578 

916 

402 

649 

197 

378 

961 

103 

695 

44 

17 

208 

575 

945 

399 

678 

193 

407 

957 

132 

690 

43 

18 

237 

572 

.10973 

396 

706 

189 

436 

953 

160 

686 

42 

19 

266 

570 

.11002 

393 

735 

186 

464 

948 

189 

681 

41 

20 

.09295 

.99567 

.11031 

.99390 

.12764 

.99182 

.14493 

.98944 

.16218 

.98676 

40 

21 

324 

564 

060 

386 

793 

178 

522 

940 

246 

671 

39 

22 

353 

562 

089 

383 

822 

175 

551 

936 

275 

667 

38 

23 

382 

559 

118 

380 

851 

171 

580 

931 

304 

662 

37 

24 

411 

556 

147 

377 

880 

167 

608 

927 

333 

657 

36 

25 

.09440 

.99553 

.11176 

.99374 

.12908 

.99163 

.14637 

.98923 

.16361 

.98652 

35 

26 

469 

551 

205 

370 

937 

160 

666 

919 

390 

648 

34 

27 

498 

548 

234 

367 

966 

156 

695 

914 

419 

643 

33 

28 

527 

545 

263 

364 

.12995 

152 

723 

910 

447 

638 

32 

29 

556 

542 

291 

360 

.13024 

148 

752 

906 

476 

633 

31 

30 

.09585 

.99540 

.11320 

.99357 

.13053 

.99144 

.14781 

.98902 

.16505 

.98629 

30 

31 

614 

537 

349 

354 

081 

141 

810 

897 

533 

624 

29 

32 

642 

534 

378 

351 

110 

137 

838 

893 

562 

619 

28 

33 

671 

531 

407 

347 

139 

133 

867 

889 

591 

614 

27 

34 

700 

528 

436 

344 

168 

129 

896 

884 

620 

609 

26 

35 

.09729 

.99526 

.11465 

.99341 

.13197 

.99125 

.14925 

.98880 

.16648 

.98604 

25 

36 

758 

523 

494 

337 

226 

122 

954 

876 

677 

600 

24 

37 

787 

520 

523 

334 

254 

118 

.14982 

871 

706 

595 

23 

38 

816 

517 

552 

331 

283 

114 

.15011 

867 

734 

590 

22 

39 

845 

514 

580 

327 

312 

110 

040 

863 

763 

585 

21 

40 

.09874 

.99511 

.11609 

.99324 

.13341 

.99106 

.15069 

.98858 

.16792 

.98580 

20 

41 

903 

508 

638 

320 

370 

102 

097 

854 

820 

575 

19 

42 

932 

506 

667 

317 

399 

098 

126 

849 

849 

570 

18 

43 

961 

503 

696 

314 

427 

094 

155 

845 

878 

565 

17 

44 

.09990 

500 

725 

310 

456 

091 

184 

841 

906 

561 

16 

45 

.10019 

.99497 

.11754 

.99307 

.13485 

.99087 

.15212 

.98836 

.16935 

.98556 

15 

46 

048 

494 

783 

303 

514 

083 

241 

832 

964 

551 

14 

47 

077 

491 

812 

300 

543 

079 

270 

827 

.16992 

546 

13 

48 

106 

488 

840 

297 

572 

075 

299 

823 

.17021 

541 

12 

49 

135 

485 

869 

293 

600 

071 

327 

818 

050 

536 

11 

50 

.10164 

.99482 

.11898 

.99290 

.13629 

.99067 

.15356 

.98814 

.17078 

.98531 

10 

51 

192 

479 

927 

286 

658 

063 

385 

809 

107 

526 

9 

52 

221 

476 

956 

283 

687 

059 

414 

805 

136 

521 

8 

53 

250 

473 

.11985 

279 

716 

055 

442 

800 

164 

516 

7 

54 

279 

470 

.12014 

276 

744 

051 

471 

796 

193 

511 

6 

55 

.10308 

.99467 

.12043 

.99272 

.13773 

.99047 

.15500 

.98791 

.17222 

.98506 

5 

56 

337 

464 

071 

269 

802 

043 

529 

787 

250 

501 

4 

57 

366 

461 

100 

265 

831 

039 

557 

782 

279 

496 

3 

58 

395 

458 

129 

262 

860 

035 

586 

778 

308 

491 

2 

59 

424 

455 

158 

258 

889 

031 

615 

773 

336 

486 

1 

60 

.10453 

.99452 

.12187 

.99255 

.13917 

.99027 

.15643 

.98769 

.17365 

.98481 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


o 

00 

83° 

o 

e* 

00 

o 

tH 

00 

00 

o 

o 



[ 90 ] 





























































































39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


PLACE VALUES: TANGENT AND COTANGENT 


6 ° 


Tan 


.10510 

540 

569 

599 

628 

.10657 

687 

716 

746 

775 

.10805 


834 

863 

893 

922 

10952 

10981 

11011 

040 

070 

11099 


Cot 


9.5144 

.4878 

.4614 

.4352 

.4090 

9.3831 

.3572 

.3315 

.3060 

.2806 

9.2553 


128 

158 

187 

217 

11246 

276 

305 

335 

364 

11394 


423 

452 

482 

511 

L 1541 

570 

600 

629 

659 

.1688 


718 

747 

777 

806 

1836 

865 

895 

924 

954 

1983 


2013 

042 

072 

101 

2131 

160 

190 

219 

249 

2278 


Zot 


.2302 

.2052 

.1803 

.1555 

9.1309 

.1065 

.0821 

.0579 

.0338 

9.0098 


8.9860 

.9623 

.9387 

.9152 

8.8919 

.8686 

.8455 

.8225 

.7996 

8.7769 


.7542 

.7317 

.7093 

.6870 

8.6648 

.6427 

.6208 

.5989 

.5772 

8.5555 


.5340 

.5126 

.4913 

.4701 

8.4490 

.4280 

.4071 

.3863 

.3656 

8.3450 


.3245 

.3041 

.2838 

.2636 

8.2434 

.2234 

.2035 

.1837 

.1640 

8.1443 


Tan 


Tan 


.12278 

308 

338 

367 

397 

.12426 

456 

485 

515 

544 

.12574 


Cot 


8.1443 


603 

633 

662 

692 

.12722 

751 

781 

810 

840 

.12869 


899 

929 

958 

.12988 

.13017 

047 

076 

106 

136 

.13165 


195 

224 

254 

284 

.13313 

343 

372 

402 

432 

.13461 


491 

521 

550 

580 

.13609 

639 

669 

698 

728 

.13758 


787 

817 

846 

876 

.13906 

935 

965 

.13995 

.14024 

.14054 


Cot 


.5618 

.5449 

.5281 

7.5113 

.4947 

.4781 

.4615 

.4451 

7.4287 


.4124 

.3962 

.3800 

.3639 

7.3479 

.3319 

.3160 

.3002 

.2844 

7.2687 


.2531 

.2375 

.2220 

.2066 

7.1912 

.1759 

.1607 

.1455 

.1304 

7.1154 


Tan 


83° 


82° 


8° 

Tan 

Cot 

:3 .14054 

7.1154 

8 084 

.1004 

4 113 

.0855 

0 143 

.0706 

7 173 

.0558 

6 .14202 

7.0410 

5 232 

.0264 

5 262 

7.0117 

6 291 

6.9972 

8 321 

.9827 

0 .14351 

6.9682 

4 381 

.9538 

5 410 

.9395 

3 440 

.9252 

1 470 

.9110 

3 .14499 

6.8969 

1 529 

.8828 

3 559 

.8687 

3 588 

.8548 

3 618 

.8408 

l .14648 

6.8269 

5 678 

.8131 

> 707 

.7994 

737 

.7856 

767 

.7720 

.14796 

6.7584 

826 

.7448 

856 

.7313 

886 

.7179 

915 

.7045 

.14945 

6.6912 

.14975 

.6779 

.15005 

.6646 

034 

.6514 

064 

.6383 

.15094 

6.6252 

124 

.6122 

153 

.5992 

183 

.5863 

213 

.5734 

.15243 

6.5606 

272 

.5478 

302 

.5350 

332 

.5223 

362 

.5097 

.15391 

6.4971 . 

421 

.4846 

451 

.4721 

481 

.4596 

511 

.4472 

.15540 

6.4348 . 

570 

.4225 

600 

.4103 

630 

.3980 

660 

.3859 

.15689 

6.3737 .] 

719 

.3617 

749 

.3496 

779 

.3376 

809 

.3257 

.15838 

6.3138 .1 

Cot 

Tan 

l 

o 

rH 

00 


Tan 


.15838 

868 

898 

928 

958 

.15988 

.16017 

047 

077 

107 

.16137 


Cot 


6.3138 60 


167 

196 

226 

256 

.16286 

316 

346 

376 

405 

.16435 


465 

495 

525 

555 

.16585 

615 

645 

674 

704 

.16734 


764 

794 

824 

854 

.16884 

914 

944 

.16974 

.17004 

.17033 


.0188 34 
6.0080 33 
5.9972 32 
.9865 31 
5.9758 30 


063 
093 
123 
153 
.17183 

213 
243 
273 
303 
.17333 


363 

393 

423 

453 

7483 

513 

543 

573 

603 


Cot 


.7594 

.7495 

.7396 

.7297 

5.7199 

.7101 

.7004 

.6906 

.6809 

5.6713 


Tan 


80° 


[ 91 ] 




















































































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



o 

O 

▼H 

11° 

12° 

13° 

14° 


/ 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

— 


0 

.17365 

.98481 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

393 

476 

109 

157 

820 

809 

523 

430 

220 

023 

59 

2 

422 

471 

138 

152 

848 

803 

552 

424 

249 

015 

58 

3 

451 

466 

167 

146 

877 

797 

580 

417 

277 

008 

57 

4 

479 

461 

195 

140 

905 

791 

608 

411 

305 

.97001 

56 

5 

.17508 

.98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

6 

537 

450 

252 

129 

962 

778 

665 

398 

362 

987 

54 

7 

565 

445 

281 

124 

.20990 

772 

693 

391 

390 

980 

53 

8 

594 

440 

309 

118 

.21019 

766 

722 

384 

418 

973 

52 

9 

623 

435 

338 

112 

047 

760 

750 

378 

446 

966 

51 

10 

,17651 

.98430 

.19366 

.98107 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

680 

425 

395 

101 

104 

748 

807 

365 

503 

952 

49 

12 

708 

420 

423 

096 

132 

742 

835 

358 

531 

945 

48 

13 

737 

414 

452 

090 

161 

735 

863 

351 

559 

937 

47 

14 

766 

409 

481 

084 

189 

729 

892 

345 

587 

930 

46 

15 

.17794 

.98404 

.19509 

.98079 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

823 

399 

538 

073 

246 

717 

948 

331 

644 

916 

44 

17 

852 

394 

566 

067 

275 

711 

.22977 

325 

672 

909 

43 

18 

880 

389 

595 

061 

303 

705 

.23005 

318 

700 

902 

42 

19 

909 

383 

623 

056 

331 

698 

033 

311 

728 

894 

41 

20 

.17937 

.98378 

.19652 

.98050 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

966 

373 

680 

044 

388 

686 

090 

298 

784 

880 

39 

22 

.17995 

368 

709 

039 

417 

680 

118 

291 

813 

873 

38 

23 

.18023 

362 

737 

033 

445 

673 

146 

284 

841 

866 

37 

24 

052 

357 

766 

027 

474 

667 

175 

278 

869 

858 

36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661 

.23203 

.97271 

.24897 

.96851 

35 

26 

109 

347 

823 

016 

530 

655 

231 

264 

925 

844 

34 

27 

138 

341 

851 

010 

559 

648 

260 

257 

954 

837 

33 

28 

166 

336 

880 

.98004 

587 

642 

288 

251 

.24982 

829 

32 

29 

195 

331 

908 

.97998 

616 

636 

316 

244 

.25010 

822 

31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630 

.23345 

.97237 

.25038 

.96815 

30 

31 

252 

320 

965 

987 

672 

623 

373 

230 

066 

807 

29 

32 

281 

315 

.19994 

981 

701 

617 

401 

223 

094 

800 

28 

33 

309 

310 

.20022 

975 

729 

611 

429 

217 

122 

793 

27 

34 

338 

304 

051 

969 

758 

604 

458 

210 

151 

786 

26 

35 

.18367 

.98299 

.20079 

.97963 

.21786 

.97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

395 

294 

108 

958 

814 

592 

514 

196 

207 

771 

24 

37 

424 

288 

136 

952 

843 

585 

542 

189 

235 

764 

23 

38 

452 

283 

165 

946 

871 

579 

571 

182 

263 

756 

22 

39 

481 

277 

193 

940 

899 

573 

599 

176 

291 

749 

21 

40 

.18509 

.98272 

.20222 

.97934 

.21928 

.97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

538 

267 

250 

928 

956 

560 

656 

162 

348 

734 

19 

42 

567 

261 

279 

922 

.21985 

553 

684 

155 

376 

727 

18 

43 

595 

256 

307 

916 

.22013 

547 

712 

148 

404 

719 

17 

44 

624 

250 

336 

910 

041 

541 

740 

141 

432 

712 

16 

45 

.18652 

.98245 

.20364 

.97905 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705 

15 

46 

681 

240 

393 

899 

098 

528 

797 

127 

488 

697 

14 

47 

710 

234 

421 

893 

126 

521 

825 

120 

516 

690 

13 

48 

738 

229 

450 

887 

155 

515 

853 

113 

545 

682 

12 

49 

767 

223 

478 

881 

183 

508 

882 

106 

573 

675 

11 

50 

.18795 

.98218 

.20507 

.97875 

.22212 

.97502 

.23910 

.97100 

.25601 

.96667 

10 

51 

824 

212 

535 

869 

240 

496 

938 

093 

629 

660 

9 

52 

852 

207 

563 

863 

268 

489 

966 

086 

657 

653 

8 

53 

881 

201 

592 

857 

297 

483 

.23995 

079 

685 

645 

7 

54 

910 

196 

620 

851 

325 

476 

.24023 

072 

713 

638 

6 

55 

.18938 

.98190 

.20649 

.97845 

.22353 

.97470 

.24051 

.97065 

.25741 

.96630 

5 

56 

967 

185 

677 

839 

382 

463 

079 

058 

769 

623 

4 

57 

.18995 

179 

706 

833 

410 

457 

108 

051 

798 

615 

3 

58 

.19024 

174 

734 

827 

438 

450 

136 

044 

826 

608 

2 

59 

052 

168 

763 

821 

467 

444 

164 

037 

854 

600 

1 

60 

.19081 

.98163 

.20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882 

.96593 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 



79° 

o 

co 

c** 

77° 

76° 

75° 



[ 92 ] 



































































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



o 

O 

▼H 

11° 

12° 

13° 

14° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

.17633 

5.6713 

.19438 

5.1446 

.21256 

4.7046 

.23087 

4.3315 

.24933 

4.0108 

60 

1 

663 

.6617 

468 

.1366 

286 

.6979 

117 

.3257 

964 

0058 

59 

! 2 

693 

.6521 

498 

.1286 

316 

.6912 

148 

.3200 

.24995 

4.0009 

58 

3 

723 

.6425 

529 

.1207 

347 

.6845 

179 

.3143 

.25026 

3.9959 

57 

4 

7 53 

.6329 

559 

: 1128 

377 

.6779 

209 

.3086 

056 

9910 

56 

5 

.17783 

5.6234 

.19589 

5.1049 

.21408 

4.6712 

.23240 

4.3029 

.25087 

3.9861 

55 

6 

813 

.6140 

619 

.0970 

438 

.6646 

271 

.2972 

118 

.9812 

54 

7 

843 

.6045 

649 

.0892 

469 

.6580 

301 

.2916 

149 

.9763 

53 

8 

873 

.5951 

680 

.0814 

499 

.6514 

332 

.2859 

180 

.9714 

52 

i 9 

903 

.5857 

710 

.0736 

529 

.6448 

363 

.2803 

211 

9665 

51 

10 

.17933 

5.5764 

.19740 

5.0658 

.21560 

4.6382 

.23393 

4.2747 

.25242 

3.9617 

50 

11 

963 

.5671 

770 

.0581 

590 

.6317 

424 

.2691 

273 

.9568 

49 

12 

.17993 

.5578 

801 

.0504 

621 

.6252 

455 

.2635 

304 

.9520 

48 

13 

.18023 

.5485 

831 

.0427 

651 

.6187 

485 

.2580 

335 

.9471 

47 

14 

053 

.5393 

861 

.0350 

682 

.6122 

516 

.2524 

366 

.9423 

46 

15 

.18083 

5.5301 

.19891 

5.0273 

.21712 

4.6057 

.23547 

4.2468 

.25397 

3.9375 

45 

16 

113 

.5209 

921 

.0197 

743 

.5993 

578 

.2413 

428 

.9327 

44 

17 

143 

.5118 

952 

.0121 

773 

.5928 

608 

.2358 

459 

.9279 

43 

18 

173 

.5026 

.19982 

5.0045 

804 

.5864 

639 

.2303 

490 

.9232 

42 

19 

203 

.4936 

.20012 

4.9969 

834 

.5800 

670 

.2248 

521 

.9184 

41 

20 

.18233 

5.4845 

.20042 

4.9894 

.21864 

4.5736 

.23700 

4.2193 

.25552 

3.9136 

40 

21 

263 

.4755 

073 

.9819 

895 

.5673 

731 

.2139 

583 

.9089 

39 

22 

293 

.4665 

103 

.9744 

925 

.5609 

762 

.2084 

614 

.9042 

38 

23 

323 

.4575 

133 

.9669 

956 

.5546 

793 

.2030 

645 

.8995 

37 

24 

353 

.4486 

164 

.9594 

.21986 

.5483 

823 

.1976 

676 

.8947 

36 

25 

.18384 

5.4397 

.20194 

4.9520 

.22017 

4.5420 

.23854 

4.1922 

.25707 

3.8900 

35 

26 

414 

.4308 

224 

.9446 

047 

.5357 

885 

.1868 

738 

.8854 

34 

27 

444 

.4219 

254 

.9372 

078 

.5294 

916 

.1814 

769 

.8807 

33 

28 

474 

.4131 

285 

.9298 

108 

.5232 

946 

.1760 

800 

.8760 

32 

29 

504 

.4043 

315 

.9225 

139 

.5169 

.23977 

.1706 

831 

.8714 

31 

30 

.18534 

5.3955 

.20345 

4.9152 

.22169 

4.5107 

.24008 

4.1653 

.25862 

3.8667 

30 

31 

564 

.3868 

376 

.9078 

200 

.5045 

039 

.1600 

893 

.8621 

29 

32 

594 

.3781 

406 

.9006 

231 

.4983 

069 

.1547 

924 

.8575 

28 

33 

624 

.3694 

436 

.8933 

261 

.4922 

100 

.1493 

955 

.8528 

27 

34 

654 

.3607 

466 

.8860 

292 

.4860 

131 

.1441 

.25986 

.8482 

26 

35 

.18684 

5.3521 

.20497 

4.8788 

.22322 

4.4799 

.24162 

4.1388 

.26017 

3.8436 

25 

36 

714 

.3435 

527 

.8716 

353 

.4737 

193 

.1335 

048 

.8391 

24 

37 

745 

.3349 

557 

.8644 

383 

.4676 

223 

.1282 

079 

.8345 

23 

38 

775 

.3263 

588 

.8573 

414 

.4615 

254 

.1230 

110 

.8299 

22 

39 

805 

.3178 

618 

.8501 

444 

.4555 

285 

.1178 

141 

.8254 

21 

40 

.18835 

5.3093 

.20648 

4.8430 

.22475 

4.4494 

.24316 

4.1126 

.26172 

3.8208 

20 

41 

865 

.3008 

679 

.8359 

505 

.4434 

347 

.1074 

203 

.8163 

19 

42 

895 

.2924 

709 

.8288 

536 

.4373 

377 

.1022 

235 

.8118 

18 

43 

925 

.2839 

739 

.8218 

567 

.4313 

408 

.0970 

266 

.8073 

17 

44 

955 

.2755 

770 

.8147 

597 

.4253 

439 

.0918 

297 

.8028 

16 

45 

.18986 

5.2672 

.20800 

4.8077 

.22628 

4.4194 

.24470 

4.0867 

.26328 

3.7983 

15 

46 

.19016 

.2588 

830 

.8007 

658 

.4134 

501 

.0815 

359 

.7938 

14 

47 

046 

.2505 

861 

.7937 

689 

.4075 

532 

.0764 

390 

.7893 

13 

48 

076 

.2422 

891 

.7867 

719 

.4015 

562 

.0713 

421 

.7848 

12 

49 

106 

.2339 

921 

.7798 

750 

.3956 

593 

.0662 

452 

.7804 

11 

50 

.19136 

5.2257 

.20952 

4.7729 

.22781 

4.3897 

.24624 

4.0611 

.26483 

3.7760 

10 

51 

166 

.2174 

.20982 

.7659 

811 

.3838 

655 

.0560 

515 

.7715 

9 

52 

197 

.2092 

.21013 

.7591 

842 

.3779 

686 

.0509 

546 

.7671 

8 

53 

227 

.2011 

043 

.7522 

872 

.3721 

717 

.0459 

577 

.7627 

7 

54 

257 

.1929 

073 

.7453 

903 

.3662 

747 

.0408 

608 

.7583 

6 

55 

.19287 

5.1848 

.21104 

4.7385 

.22934 

4.3604 

.24778 

4.0358 

.26639 

3.7539 

5 

56 

317 

.1767 

134 

.7317 

964 

.3546 

809 

.0308 

670 

.7495 

4 

57 

347 

.1686 

164 

.7249 

.22995 

.3488 

840 

.0257 

701 

.7451 

3 

58 

378 

.1606 

195 

.7181 

.23026 

.3430 

871 

.0207 

733 

.7408 

2 

59 

408 

.1526 

225 

.7114 

056 

.3372 

902 

.0158 

764 

.7364 

1 

60 

.19438 

5.1446 

.21256 

4.7046 

.23087 

4.3315 

.24933 

4.0108 

.26795 

3.7321 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 


79° 

o 

00 

77° 

o 

O 

75° 



[ 93 ] 














































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



15° 

16° 

17° 

o 

oo 

rH 

19° 



Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.25882 

.96593 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

.32557 

.94552 

60 

1 

910 

585 

592 

118 

265 

622 

929 

097 

584 

542 

59 

2 

938 

578 

620 

110 

293 

613 

957 

088 

612 

533 

58 

3 

966 

570 

648 

102 

321 

605 

.30985 

079 

639 

523 

57 

4 

.25994 

562 

676 

094 

348 

596 

.31012 

070 

667 

514 

56 

5 

.26022 

.96555 

.27704 

.96086 

.29376 

.95588 

.31040 

.95061 

.32694 

.94504 

55 

6 

050 

547 

731 

078 

404 

579 

068 

052 

722 

495 

54 

7 

079 

540 

759 

070 

432 

571 

095 

043 

749 

485 

53 

8 

107 

532 

787 

062 

460 

562 

123 

033 

777 

476 

52 

9 

135 

524 

815 

054 

487 

554 

151 

024 

804 

466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

191 

509 

871 

037 

543 

536 

206 

.95006 

859 

447 

49 

12 

219 

502 

899 

029 

571 

528 

233 

.94997 

887 

438 

48 

13 

247 

494 

927 

021 

599 

519 

261 

988 

914 

428 

47 

14 

275 

486 

955 

013 

626 

511 

289 

979 

942 

418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316 

.94970 

.32969 

.94409 

45 

16 

331 

471 

.28011 

.95997 

682 

493 

344 

961 

.32997 

399 

44 

17 

359 

463 

039 

989 

710 

485 

372 

952 

.33024 

390 

43 

18 

387 

456 

067 

981 

737 

476 

399 

943 

051 

380 

42 

19 

415 

448 

095 

972 

765 

467 

427 

933 

079 

370 

41 

20 

.26443 

.96440 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

471 

433 

150 

956 

821 

450 

482 

915 

134 

351 

39 

22 

500 

425 

178 

948 

849 

441 

510 

906 

161 

342 

38 

23 

528 

417 

206 

940 

876 

433 

537 

897 

189 

332 

37 

24 

556 

410 

234 

931 

904 

424 

565 

888 

216 

322 

36 

25 

.26584 

.96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

612 

394 

290 

915 

960 

407 

620 

869 

271 

303 

34 

27 

640 

386 

318 

907 

.29987 

398 

648 

860 

298 

293 

33 

28 

668 

379 

346 

898 

.30015 

389 

675 

851 

326 

284 

32 

29 

696 

371 

374 

890 

043 

380 

703 

842 

353 

274 

31 

30 

.26724 

.96363 

.28402 

.95882 

.30071 

.95372 

.31730 

.94832 

.33381 

.94264 

30 

31 

752 

355 

429 

874 

098 

363 

758 

823 

408 

254 

29 

32 

780 

347 

457 

865 

126 

354 

786 

814 

436 

245 

28 

33 

808 

340 

485 

857 

154 

345 

813 

805 

463 

235 

27 

34 

836 

332 

513 

849 

182 

337 

841 

795 

490 

225 

26 

35 

126864 

.96324 

.28541 

.95841 

.30209 

.95328 

.31868 

.94786 

.33518 

.94215 

25 

36 

892 

316 

569 

832 

237 

319 

896 

777 

545 

206 

24 

37 

920 

308 

597 

824 

265 

310 

923 

768 

573 

196 

23 

38 

948 

301 

625 

816 

292 

301 

951 

758 

600 

186 

22 

39 

.26976 

293 

652 

807 

320 

293 

.31979 

749 

627 

176 

21 

40 

.27004 

.96285 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

032 

277 

708 

791 

376 

275 

034 

730 

682 

157 

19 

42 

060 

269 

736 

782 

403 

266 

061 

721 

710 

147 

18 

43 

088 

261 

764 

774 

431 

257 

089 

712 

737 

137 

17 

44 

116 

253 

792 

766 

459 

248 

116 

702 

764 

127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

172 

238 

847 

749 

514 

231 

171 

684 

819 

108 

14 

47 

200 

230 

875 

740 

542 

222 

199 

674 

846 

098 

13 

48 

228 

222 

903 

732 

570 

213 

227 

665 

874 

088 

12 

49 

256 

214 

931 

724 

597 

204 

254 

656 

901 

078 

11 

50 

.27284 

.96206 

.28959 

.95715 

.30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

312 

198 

.28987 

707 

653 

186 

309 

637 

956 

058 

9 

52 

340 

190 

.29015 

698 

680 

177 

337 

627 

.33983 

049 

8 

53 

368 

182 

042 

690 

708 

168 

364 

618 

.34011 

039 

7 

54 

396 

174 

070 

681 

736 

159 

392 

609 

038 

029 

6 

55 

.27424 

.96166 

.29098 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

.94019 

5 

56 

452 

158 

126 

664 

791 

142 

447 

590 

093 

.94009 

4 

57 

480 

150 

154 

656 

819 

133 

474 

580 

120 

.93999 

3 

58 

508 

142 

182 

647 

846 

124 

502 

571 

147 

989 

2 

59 

536 

134 

209 

639 

874 

115 

529 

561 

175 

979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.95106 

32557 

.94552 

.34202 

.93969 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


74° 

73° 

72° 

71° 

o 

O 



[ 94 ] 





















































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



15° 

16° 

17° 

o 

00 

rH 

19° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

.26795 

3.7321 

.28675 

3.4874 

.30573 

3.2709 

.32492 

3.0777 

.34433 

2.9042 

60 

1 

826 

.7277 

706 

.4836 

605 

.2675 

524 

.0746 

465 

.9015 

59 

2 

857 

.7234 

738 

.4798 

637 

.2641 

556 

.0716 

498 

.8987 

58 

3 

888 

.7191 

769 

.4760 

669 

.2607 

588 

.0686 

530 

8960 

57 

4 

920 

.7148 

801 

.4722 

700 

.2573 

621 

.0655 

563 

.8933 

56 

5 

.26951 

3.7105 

.28832 

3.4684 

.30732 

3.2539 

.32653 

3.0625 

.34596 

2.8905 

55 

6 

.26982 

.7062 

864 

.4646 

764 

.2506 

685 

.0595 

628 

.8878 

54 

7 

.27013 

.7019 

895 

.4608 

796 

.2472 

717 

.0565 

661 

.8851 

53 

8 

044 

.6976 

927 

.4570 

828 

.2438 

749 

.0535 

693 

.8824 

52 

9 

076 

.6933 

958 

.4533 

860 

.2405 

782 

.0505 

726 

.8797 

51 

10 

.27107 

3.6891 

.28990 

3.4495 

.30891 

3.2371 

.32814 

3.0475 

.34758 

2.8770 

50 

11 

138 

.6848 

.29021 

.4458 

923 

.2338 

846 

.0445 

791 

.8743 

49 

12 

169 

.6806 

053 

.4420 

955 

.2305 

878 

.0415 

824 

.8716 

48 

13 

201 

.6764 

084 

.4383 

.30987 

.2272 

911 

.0385 

856 

.8689 

47 

14 

232 

.6722 

116 

.4346 

.31019 

.2238 

943 

.0356 

889 

.8662 

46 

15 

.27263 

3.6680 

.29147 

3.4308 

.31051 

3.2205 

.32975 

3.0326 

.34922 

2.8636 

45 

16 

294 

.6638 

179 

.4271 

083 

.2172 

.33007 

.0296 

954 

.8609 

44 

17 

326 

.6596 

210 

.4234 

115 

.2139 

040 

.0267 

.34987 

.8582 

43 

18 

357 

.6554 

242 

.4197 

147 

.2106 

072 

.0237 

.35020 

.8556 

42 

19 

388 

.6512 

274 

.4160 

178 

.2073 

104 

.0208 

052 

.8529 

41 

20 

.27419 

3.6470 

.29305 

3.4124 

.31210 

3.2041 

.33136 

3.0178 

.35085 

2.8502 

40 

21 

451 

.6429 

337 

.4087 

242 

.2008 

169 

.0149 

118 

.8476 

39 

22 

482 

.6387 

368 

.4050 

274 

.1975 

201 

.0120 

150 

.8449 

38 

23 

513 

.6346 

400 

.4014 

306 

.1943 

233 

.0090 

183 

.8423 

37 

24 

545 

.6305 

432 

.3977 

338 

.1910 

266 

.0061 

216 

.8397 

36 

25 

.27576 

3.6264 

.29463 

3.3941 

.31370 

3.1878 

.33298 

3.0032 

.35248 

2.8370 

35 

26 

607 

.6222 

495 

.3904 

402 

.1845 

330 

3.0003 

281 

.8344 

34 

27 

638 

.6181 

526 

.3868 

434 

.1813 

363 

2.9974 

314 

.8318 

33 

28 

670 

.6140 

558 

.3832 

466 

.1780 

395 

.9945 

346 

.8291 

32 

29 

701 

.6100 

590 

.3796 

498 

.1748 

427 

.9916 

379 

.8265 

31 

30 

.27732 

3.6059 

.29621 

3.3759 

.31530 

3.1716 

.33460 

2.9887 

.35412 

2.8239 

30 

31 

764 

.6018 

653 

.3723 

562 

.1684 

492 

.9858 

445 

.8213 

29 

32 

795 

.5978 

685 

.3687 

594 

.1652 

524 

.9829 

477 

.8187 

28 

33 

826 

.5937 

716 

.3652 

626 

.1620 

557 

.9800 

510 

.8161 

27 

34 

858 

.5897 

748 

.3616 

658 

.1588 

589 

.9772 

543 

.8135 

26 

35 

.27889 

3.5856 

.29780 

3.3580 

.31690 

3.1556 

.33621 

2.9743 

.35576 

2.8109 

25 

36 

921 

.5816 

811 

.3544 

722 

.1524 

654 

.9714 

608 

.8083 

24 

37 

952 

.5776 

843 

.3509 

754 

.1492 

686 

.9686 

641 

.8057 

23 

38 

.27983 

.5736 

875 

.3473 

786 

.1460 

718 

.9657 

674 

.8032 

22 

39 

.28015 

.5696 

906 

.3438 

818 

.1429 

751 

.9629 

707 

.8006 

21 

40 

.28046 

3.5656 

.29938 

3.3402 

.31850 

3.1397 

.33783 

2.9600 

.35740 

2.7980 

20 

41 

077 

.5616 

.29970 

.3367 

882 

.1366 

816 

.9572 

772 

.7955 

19 

42 

109 

.5576 

.30001 

.3332 

914 

.1334 

848 

.9544 

805 

.7929 

18 

43 

140 

.5536 

033 

.3297 

946 

.1303 

881 

.9515 

838 

.7903 

17 

44 

172 

.5497 

065 

.3261 

.31978 

.1271 

913 

.9487 

871 

.7878 

16 

45 

.28203 

3.5457 

.30097 

3.3226 

.32010 

3.1240 

.33945 

2.9459 

.35904 

2.7852 

15 

46 

234 

.5418 

128 

.3191 

042 

.1209 

.33978 

.9431 

937 

.7827 

14 

47 

266 

.5379 

160 

.3156 

074 

.1178 

.34010 

.9403 

.35969 

.7801 

13 

48 

297 

.5339 

192 

.3122 

106 

.1146 

043 

.9375 

.36002 

.7776 

12 

49 

329 

.5300 

224 

.3087 

139 

.1115 

075 

.9347 

035 

.7751 

11 

50 

.28360 

3.5261 

.30255 

3.3052 

.32171 

3.1084 

.34108 

2.9319 

.36068 

2.7725 

10 

51 

391 

.5222 

287 

.3017 

203 

.1053 

140 

.9291 

101 

.7700 

9 

52 

423 

.5183 

319 

.2983 

235 

.1022 

173 

.9263 

134 

.7675 

8 

53 

454 

.5144 

351 

.2948 

267 

.0991 

205 

.9235 

167 

.7650 

7 

54 

486 

.5105 

382 

.2914 

299 

.0961 

238 

.9208 

199 

.7625 

6 

55 

.28517 

3.5067 

.30414 

3.2879 

.32331 

3.0930 

.34270 

2.9180 

.36232 

2.7600 

5 

56 

549 

.5028 

446 

.2845 

363 

.0899 

303 

.9152 

265 

.7575 

4 

57 

580 

.4989 

478 

.2811 

396 

.0868 

335 

.9125 

298 

.7550 

3 

58 

612 

.4951 

509 

.2777 

428 

.0838 

368 

.9097 

331 

.7525 

2 

59 

643 

.4912 

541 

.2743 

460 

.0807 

400 

.9070 

364 

.7500 

1 

60 

.28675 

3.4874 

.30573 

3.2709 

.32492 

3.0777 

.34433 

2.9042 

.36397 

2.7475 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 


74° 

73 

o 

72 

o 

71 

o 

70 c 




[ 95 ] 
















































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 


- 

20° 

21° 

22° 

23° 

24° 



Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.34202 

.93969 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

60 

1 

229 

959 

864 

348 

488 

707 

100 

039 

700 

343 

59 

2 

257 

949 

891 

337 

515 

697 

127 

028 

727 

331 

58 

3 

284 

939 

918 

327 

542 

686 

153 

016 

753 

319 

57 

4 

311 

929 

945 

316 

569 

675 

180 

.92005 

780 

307 

56 

5 

.34339 

.93919 

.35973 

.93306 

.37595 

.92664 

.39207 

.91994 

.40806 

.91295 

55 

6 

366 

909 

.36000 

295 

622 

653 

234 

982 

833 

283 

54 

7 

393 

899 

027 

285 

649 

642 

260 

971 

860 

272 

53 

8 

421 

889 

054 

274 

676 

631 

287 

959 

886 

260 

52 

9 

448 

879 

081 

264 

703 

620 

314 

948 

913 

248 

51 

10 

.34475 

.93869 

.36108 

.93253 

.37730 

.92609 

.39341 

.91936 

.40939 

.91236 

50 

11 

503 

859 

135 

243 

757 

598 

367 

925 

966 

224 

49 

12 

530 

849 

162 

232 

784 

587 

394 

914 

.40992 

212 

48 

13 

557 

839 

190 

222 

811 

576 

421 

902 

.41019 

200 

47 

14 

584 

829 

217 

211 

838 

565 

448 

891 

045 

188 

46 

15 

.34612 

.93819 

.36244 

.93201 

.37865 

.92554 

.39474 

.91879 

.41072 

.91176 

45 

16 

639 

809 

271 

190 

892 

543 

501 

868 

098 

164 

44 

17 

666 

799 

298 

180 

919 

532 

528 

856 

125 

152 

43 

18 

694 

789 

325 

169 

946 

521 

555 

845 

151 

140 

42 

19 

721 

779 

352 

159 

973 

510 

581 

833 

178 

128 

41 

20 

.34748 

.93769 

.36379 

.93148 

.37999 

.92499 

.39608 

.91822 

.41204 

.91116 

40 

21 

775 

759 

406 

137 

.38026 

488 

635 

810 

231 

104 

39 

22 

803 

748 

434 

127 

053 

477 

661 

799 

257 

092 

38 

23 

830 

738 

461 

116 

080 

466 

688 

787 

284 

080 

37 

24 

857 

728 

488 

106 

107 

455 

715 

775 

310 

068 

36 

25 

.34884 

.93718 

.36515 

.93095 

.38134 

.92444 

.39741 

.91764 

.41337 

.91056 

35 

26 

912 

708 

542 

084 

161 

432 

768 

752 

363 

044 

34 

27 

939 

698 

569 

074 

188 

421 

795 

741 

390 

032 

33 

28 

966 

688 

596 

063 

215 

410 

822 

729 

416 

020 

32 

29 

.34993 

677 

623 

052 

241 

399 

848 

718 

443 

.91008 

31 

30 

.35021 

.93667 

.36650 

.93042 

.38268 

.92388 

.39875 

.91706 

.41469 

.90996 

30 

31 

048 

657 

677 

031 

295 

377 

902 

694 

496 

984 

29 

32 

075 

647 

704 

020 

322 

366 

928 

683 

522 

972 

28 

33 

102 

637 

731 

.93010 

349 

355 

955 

671 

549 

960 

27 

34 

130 

626 

758 

.92999 

376 

343* 

.39982 

660 

575 

948 

26 

35 

.35157 

.93616 

.36785 

.92988 

.38403 

.92332 

.40008 

.91648 

.41602 

.90936 

25 

36 

184 

606 

812 

978 

430 

321 

035 

636 

628 

924 

24 

37 

211 

596 

839 

967 

456 

310 

062 

625 

655 

911 

23 

38 

239 

585 

867 

956 

483 

299 

088 

613 

681 

899 

22 

39 

266 

575 

894 

945 

510 

287 

115 

601 

707 

887 

21 

40 

.35293 

.93565 

.36921 

.92935 

.38537 

.92276 

.40141 

.91590 

.41734 

.90875 

20 

41 

320 

555 

948 

924 

564 

265 

168 

578 

760 

863 

19 

42 

347 

544 

.36975 

913 

591 

254 

195 

566 

787 

851 

18 

43 

375 

534 

.37002 

902 

617 

243 

221 

555 

813 

839 

17 

44 

402 

524 

029 

892 

644 

231 

248 

543 

840 

826 

16 

45 

.35429 

.93514 

.37056 

.92881 

.38671 

.92220 

.40275 

.91531 

.41866 

.90814 

15 

46 

456 

503 

083 

870 

698 

209 

301 

519 

892 

802 

14 

47 

484 

493 

110 

859 

725 

198 

328 

508 

919 

790 

13 

48 

511 

483 

137 

849 

752 

186 

355 

496 

945 

778 

12 

49 

538 

472 

164 

838 

778 

175 

381 

484 

972 

766 

11 

50 

.35565 

.93462 

.37191 

.92827 

.38805 

.92164 

.40408 

.91472 

.41998 

.90753 

10 

51 

592 

452 

218 

816 

832 

152 

434 

461 

.42024 

741 

9 

52 

619 

441 

245 

805 

859 

141 

461 

449 

051 

729 

8 

53 

647 

431 

272 

794 

886 

130 

488 

437 

077 

717 

7 

54 

674 

420 

299 

784 

912 

119 

514 

425 

104 

704 

6 

55 

.35701 

.93410 

.37326 

.92773 

.38939 

.92107 

.40541 

.91414 

.42130 

.90692 

5 

56 

728 

400 

353 

762 

966 

096 

567 

402 

156 

680 

4 

57 

755 

389 

380 

751 

.38993 

085 

594 

390 

183 

668 

3 

58 

782 

379 

407 

740 

.39020 

073 

621 

378 

209 

655 

2 

59 

810 

368 

434 

729 

046 

062 

647 

366 

235 

643 

1 

60 

.35837 

.93358 

.37461 

.92718 

.39073 

.92050 

.40674 

.91355 

.42262 

.90631 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


69° 

68° 

67° 

66° 

65° 



[ 96 ] 































































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



20° 

21° 

22° 

t 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

0 

.36397 

2.7475 

.38386 

2.6051 

.40403 

2.4751 

1 

430 

.7450 

420 

.6028 

436 

.4730 

2 

463 

.7425 

453 

.6006 

470 

.4709 

3 

496 

.7400 

487 

.5983 

504 

.4689 

1 4 

529 

.7376 

520 

.5961 

538 

.4668 

5 

.36562 

2.7351 

.38553 

2.5938 

.40572 

2.4648 

6 

595 

.7326 

587 

.5916 

606 

.4627 

7 

628 

.7302 

620 

.5893 

640 

.4606 

8 

661 

.7277 

654 

.5871 

674 

.4586 

9 

694 

.7253 

687 

.5848 

707 

.4566 

10 

.36727 

2.7228 

.38721 

2.5826 

.40741 

2.4545 

11 

760 

.7204 

754 

.5804 

775 

.4525 

12 

793 

.7179 

787 

.5782 

809 

.4504 

13 

826 

.7155 

821 

.5759 

843 

.4484 

14 

859 

.7130 

854 

.5737 

877 

.4464 

15 

.36892 

2.7106 

.38888 

2.5715 

.40911 

2.4443 

16 

925 

.7082 

921 

.5693 

945 

.4423 

17 

958 

.7058 

955 

.5671 

.40979 

.4403 

18 

.36991 

.7034 

.38988 

.5649 

.41013 

.4383 

19 

.37024 

.7009 

.39022 

.5627 

047 

.4362 

20 

.37057 

2.6985 

.39055 

2.5605 

.41081 

2.4342 

21 

090 

.6961 

089 

.5583 

115 

.4322 

22 

123 

.6937 

122 

.5561 

149 

.4302 

23 

157 

.6913 

156 

.5539 

183 

.4282 

24 

190 

.6889 

190 

.5517 

217 

.4262 

25 

.37223 

2.6865 

.39223 

2.5495 

.41251 

2.4242 

26 

256 

.6841 

257 

.5473 

285 

.4222 

27 

289 

.6818 

290 

.5452 

319 

.4202 

28 

322 

.6794 

324 

.5430 

353 

.4182 

29 

355 

.6770 

357 

.5408 

387 

.4162 

30 

.37388 

2.6746 

.39391 

2.5386 

.41421 

2.4142 

31 

422 

.6723 

425 

.5365 

455 

.4122 

32 

455 

.6699 

458 

.5343 

490 

.4102 

33 

488 

.6675 

492 

.5322 

524 

.4083 

34 

521 

.6652 

526 

.5300 

558 

.4063 

35 

.37554 

2.6628 

.39559 

2.5279 

.41592 

2.4043 

36 

588 

.6605 

593 

.5257 

626 

.4023 

37 

621 

.6581 

626 

.5236 

660 

.4004 

38 

654 

.6558 

660 

.5214 

694 

.3984 

39 

687 

.6534 

694 

.5193 

728 

.3964 

40 

.37720 

2.6511 

.39727 

2.5172 

.41763 

2.3945 

41 

754 

.6488 

761 

.5150 

797 

.3925 

42 

787 

.6464 

795 

.5129 

831 

.3906 

43 

820 

.6441 

829 

.5108 

865 

.3886 

44 

853 

.6418 

862 

.5086 

899 

.3867 

45 

.37887 

2.6395 

.39896 

2.5065 

.41933 

2.3847 

46 

920 

.6371 

930 

.5044 

.41968 

.3828 

47 

953 

.6348 

963 

.5023 

.42002 

.3808 

48 

.37986 

.6325 

.39997 

.5002 

036 

.3789 

49 

.38020 

.6302 

.40031 

.4981 

070 

.3770 

50 

.38053 

2.6279 

.40065 

2.4960 

.42105 

2.3750 

51 

086 

.6256 

098 

.4939 

139 

.3731 

52 

120 

.6233 

132 

.4918 

173 

.3712 

53 

153 

.6210 

166 

.4897 

207 

.3693 

54 

186 

.6187 

200 

.4876 

242 

.3673 

55 

.38220 

2.6165 

.40234 

2.4855 

.42276 

2.3654 

56 

253 

.6142 

267 

.4834 

310 

.3635 

57 

286 

.6119 

301 

.4813 

345 

.3616 

58 

320 

.6096 

335 

.4792 

379 

.3597 

59 

353 

.6074 

369 

.4772 

413 

.3578 

60 

.38386 

2.6051 

.40403 

2.4751 

.42447 

2.3559 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 


69° 

o 

00 

o 

67° 


23° 


Tan 


Cot 


.42447 

482 

516 

551 

585 

.42619 

654 

688 

722 

757 

.42791 


826 

860 

894 

929 

.42963 

.42998 

.43032 

067 

101 

.43136 


170 

205 

239 

274 

.43308 

343 

378 

412 

447 

.43481 


516 

550 

585 

620 

.43654 

689 

724 

758 

793 

.43828 


862 

897 

932 

.43966 

.44001 

036 

071 

105 

140 

.44175 


210 

244 

279 

314 

.44349 

384 

418 

453 

488 

.44523 


2.3559 


.3501 


.3351 

.3332 

.3313 

.3294 

2.3276 

.3257 

.3238 

.3220 

.3201 

2.3183 


.3164 

.3146 

.3127 

.3109 

2.3090 

.3072 

.3053 

.3035 

.3017 

2.2998 


.2980 

.2962 

.2944 

.2925 

2.2907 

.2889 

.2871 

.2853 

.2835 

2.2817 


Cot 


.2799 

.2781 

.2763 

.2745 

2.2727 

.2709 

.2691 

.2673 

.2655 

2.2637 


.2620 

.2602 

.2584 

.2566 

2.2549 

.2531 

.2513 

.2496 

.2478 

2.2460 


Tan 


66 ° 


o 

0* 


Tan 

Cot 


.44523 

2.2460 

60 

558 

.2443 

59 

593 

.2425 

58 

627 

.2408 

57 

662 

.2390 

56 

.44697 

2.2373 

55 

732 

.2355 

54 

767 

.2338 

53 

802 

.2320 

52 

837 

.2303 

51 

.44872 

2.2286 

50 

907 

.2268 

49 

942 

.2251 

48 

.44977 

.2234 

47 

.45012 

.2216 

46 

.45047 

2.2199 

45 

082 

.2182 

44 

117 

.2165 

43 

152 

.2148 

42 

187 

.2130 

41 

.45222 

2.2113 

40 

257 

.2096 

39 

292 

.2079 

38 

327 

.2062 

37 

362 

.2045 

36 

.45397 

2.2028 

35 

432 

.2011 

34 

467 

.1994 

33 

502 

.1977 

32 

538 

.1960 

31 

.45573 

2.1943 

30 

608 

.1926 

29 

643 

.1909 

28 

678 

.1892 

27 

713 

.1876 

26 

.45748 

2.1859 

25 

784 

.1842 

24 

819 

.1825 

23 

854 

.1808 

22 

889 

.1792 

21 

.45924 

2.1775 

20 

960 

.1758 

19 

.45995 

.1742 

18 

.46030 

.1725 

17 

065 

.1708 

16 

.46101 

2.1692 

15 

136 

.1675 

14 

171 

.1659 

13 

206 

.1642 

12 

242 

.1625 

11 

.46277 

2.1609 

10 

312 

.1592 

9 

348 

.1576 

8 

383 

.1560 

7 

418 

.1543 

6 

.46454 

2.1527 

5 

489 

.1510 

4 

525 

.1494 

3 

560 

.1478 

2 

595 

.1461 

1 

.46631 

2.1445 

0 

Cot 

Tan 

r 

65 

o 



[ 97 ] 



































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



25° 

26° 

27° 

28° 

29° 

1 

/ 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.42262 

.90631 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

60 

1 

288 

618 

863 

867 

425 

087 

973 

281 

506 

448 

59 

2 

315 

606 

889 

854 

451 

074 

.46999 

267 

532 

434 

58 

3 

341 

594 

916 

841 

477 

061 

.47024 

254 

557 

420 

57 

4 

367 

582 

942 

828 

503 

048 

050 

240 

583 

406 

56 

5 

.42394 

.90569 

.43968 

.89816 

.45529 

.89035 

.47076 

.88226 

.48608 

.87391 

55 

6 

420 

557 

.43994 

803 

554 

021 

101 

213 

634 

377 

54 

7 

446 

545 

.44020 

790 

580 

.89008 

127 

199 

659 

363 

53 

8 

473 

532 

046 

777 

606 

.88995 

153 

185 

684 

349 

52 

9 

499 

520 

072 

764 

632 

981 

178 

172 

710 

335 

51 

10 

.42525 

.90507 

.44098 

.89752 

.4565S 

.88968 

.47204 

.88158 

.48735 

.87321 

50 

11 

552 

495 

124 

739 

684 

955 

229 

144 

761 

306 

49 

12 

578 

483 

151 

726 

710 

942 

255 

130 

786 

292 

48 

13 

604 

470 

177 

713 

736 

928 

281 

117 

811 

278 

47 

14 

631 

458 

203 

700 

762 

915 

306 

103 

837 

264 

46 

15 

.42657 

.90446 

.44229 

.89687 

.45787 

.88902 

.47332 

.88089 

.48862 

.87250 

45| 

16 

683 

433 

255 

674 

813 

888 

358 

075 

888 

235 

44 

17 

709 

421 

281 

662 

839 

875 

383 

062 

913 

221 

43 

18 

736 

408 

307 

649 

865 

862 

409 

048 

938 

207 

42| 

19 

762 

396 

333 

636 

891 

848 

434 

034 

964 

193 

41 

20 

.42788 

.90383 

.44359 

.89623 

.45917 

.88835 

.47460 

.88020 

.48989 

.87178 

40| 

21 

815 

371 

385 

610 

942 

822 

486 

.88006 

.49014 

164 

39 

22 

841 

358 

411 

597 

968 

808 

511 

.87993 

040 

150 

38 

23 

867 

346 

437 

584 

.45994 

795 

537 

979 

065 

136 

37 

24 

894 

334 

464 

571 

.46020 

782 

562 

965 

090 

121 

36 

25 

.42920 

.90321 

.44490 

.89558 

.46046 

.88768 

.47588 

.87951 

.49116 

.87107 

35 

26 

946 

309 

516 

545 

072 

755 

614 

937 

141 

093 

34 

27 

972 

296 

542 

532 

097 

741 

639 

923 

166 

079 

33 

28 

.42999 

284 

568 

519 

123 

728 

665 

909 

192 

064 

32 

29 

.43025 

271 

594 

506 

149 

715 

690 

896 

217 

050 

31 

30 

.43051 

.90259 

.44620 

.89493 

.46175 

.88701 

.47716 

.87882 

.49242 

.87036 

30 

31 

077 

246 

646 

480 

201 

688 

741 

868 

268 

021 

29 

32 

104 

233 

672 

467 

226 

674 

767 

854 

293 

.87007 

28 

33 

130 

221 

698 

454 

252 

661 

793 

840 

318 

.86993 

27 

34 

156 

208 

724 

441 

278 

647 

818 

826 

344 

978 

26 1 

35 

.43182 

.90196 

.44750 

.89428 

.46304 

.88634 

.47844 

.87812 

.49369 

.86964 

25 

36 

209 

183 

776 

415 

330 

620 

869 

798 

394 

949 

24 

37 

235 

171 

802 

402 

355 

607 

895 

784 

419 

935 

23 1 

38 

261 

158 

828 

389 

381 

593 

920 

770 

445 

921 

22 1 

39 

287 

146 

854 

376 

407 

580 

946 

756 

470 

906 

21 1 

40 

.43313 

.90133 

.44880 

.89363 

.46433 

.88566 

.47971 

.87743 

.49495 

.86892 

20 1 

41 

340 

120 

906 

350 

458 

553 

.47997 

729 

521 

878 

19 

42 

366 

108 

932 

337 

484 

539 

.48022 

715 

546 

863 

18 1 

43 

392 

095 

958 

324 

510 

526 

048 

701 

571 

849 

17 

44 

418 

082 

.44984 

311 

536 

512 

073 

687 

596 

834 

16 

45 

.43445 

.90070 

.45010 

.89298 

.46561 

.88499 

.48099 

.87673 

.49622 

.86820 

15| 

46 

471 

057 

036 

285 

587 

485 

124 

659 

647 

805 

14 

47 

497 

045 

062 

272 

613 

472 

150 

645 

672 

791 

13 

48 

523 

032 

088 

259 

639 

458 

175 

631 

697 

777 

12 

49 

549 

019 

114 

245 

664 

445 

201 

617 

723 

762 

11 

50 

.43575 

.90007 

.45140 

.89232 

.46690 

.88431 

.48226 

.87603 

.49748 

.86748 

lol 

51 

602 

.89994 

166 

219 

716 

417 

252 

589 

773 

733 

9 

52 

628 

981 

192 

206 

742 

404 

277 

575 

798 

719 

8| 

53 

654 

968 

218 

193 

767 

390 

303 

561 

824 

704 

71 

54 

680 

956 

243 

180 

793 

377 

328 

546 

849 

690 

61 

55 

.43706 

.89943 

.45269 

.89167 

.46819 

.88363 

.48354 

.87532 

.49874 

.86675 

5| 

56 

733 

930 

295 

153 

844 

349 

379 

518 

899 

661 

4 

57 

759 

918 

321 

140 

870 

336 

405 

504 

924 

646 

3 

58 

785 

905 

347 

127 

896 

322 

430 

490 

950 

632 

21 

59 

811 

892 

373 

114 

921 

308 

456 

476 

.49975 

617 

l| 

60 

.43837 

.89879 

.45399 

.89101 

.46947 

.88295 

.48481 

.87462 

.50000 

.86603 

0 


Cos 

Sin 

Cos 1 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ I 


64° 

63 

O 

62 

0 

61 

0 

60° 

<- 


[ 98 ] 
















































































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



25° 

26° 

27° 

o 

CO 

eq 

29° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

.46631 

2.1445 

.48773 

2.0503 

.50953 

1.9626 

.53171 

1.8807 

.55431 

1.8040 

60 

1 

666 

.1429 

809 

.0488 

.50989 

.9612 

208 

.8794 

469 

.8028 

59 

I 2 

702 

.1413 

845 

.0473 

.51026 

.9598 

246 

.8781 

507 

.8016 

58 

3 

737 

.1396 

881 

.0458 

063 

.9584 

283 

.8768 

545 

.8003 

57 

4 

772 

.1380 

917 

.0443 

099 

.9570 

320 

.8755 

583 

.7991 

56 

5 

.46808 

2.1364 

.48953 

2.0428 

.51136 

1.9556 

.53358 

1.8741 

.55621 

1.7979 

55 

6 

843 

.1348 

.48989 

.0413 

173 

.9542 

395 

.8728 

659 

.7966 

54 

7 

879 

.1332 

.49026 

.0398 

209 

.9528 

432 

.8715 

697 

.7954 

53 

8 

914 

.1315 

062 

.0383 

246 

.9514 

470 

.8702 

736 

.7942 

52 

9 

950 

.1299 

098 

.0368 

283 

.9500 

507 

.8689 

774 

.7930 

51 

10 

.46985 

2.1283 

.49134 

2.0353 

.51319 

1.9486 

.53545 

1.8676 

.55812 

1.7917 

50 

11 

.47021 

.1267 

170 

.0338 

356 

.9472 

582 

.8663 

850 

.7905 

49 

12 

056 

.1251 

206 

.0323 

393 

.9458 

620 

.8650 

888 

.7893 

48 

13 

092 

.1235 

242 

.0308 

430 

.9444 

657 

.8637 

926 

.7881 

47 

14 

128 

.1219 

278 

.0293 

467 

.9430 

694 

.8624 

.55964 

.7868 

46 

15 

.47163 

2.1203 

.49315 

2.0278 

.51503 

1.9416 

.53732 

1.8611 

.56003 

1.7856 

45 

16 

199 

.1187 

351 

.0263 

540 

.9402 

769 

.8598 

041 

.7844 

44 

17 

234 

.1171 

387 

.0248 

577 

.9388 

807 

.8585 

079 

.7832 

43 

18 

270 

.1155 

423 

.0233 

614 

.9375 

844 

.8572 

117 

.7820 

42 

19 

305 

.1139 

459 

.0219 

651 

.9361 

882 

.8559 

156 

.7808 

41 

20 

.47341 

2.1123 

.49495 

2.0204 

.51688 

1.9347 

.53920 

1.8546 

.56194 

1.7796 

40 

21 

377 

.1107 

532 

.0189 

724 

.9333 

957 

.8533 

232 

.7783 

39 

22 

412 

.1092 

568 

.0174 

761 

.9319 

.53995 

.8520 

270 

.7771 

38 

23 

448 

.1076 

604 

.0160 

798 

.9306 

.54032 

.8507 

309 

.7759 

37 

24 

483 

.1060 

640 

.0145 

835 

.9292 

070 

.8495 

347 

.7747 

36 

25 

.47519 

2.1044 

.49677 

2.0130 

.51872 

1.9278 

.54107 

1.8482 

.56385 

1.7735 

35 

26 

555 

.1028 

713 

.0115 

909 

.9265 

145 

.8469 

424 

.7723 

34 

27 

590 

.1013 

749 

.0101 

946 

.9251 

183 

.8456 

462 

.7711 

33 

28 

626 

.0997 

786 

.0086 

.51983 

.9237 

220 

.8443 

501 

.7699 

32 

29 

662 

.0981 

822 

.0072 

.52020 

.9223 

258 

.8430 

539 

.7687 

31 

30 

.47698 

2.0965 

.49858 

2.0057 

.52057 

1.9210 

.54296 

1.8418 

.56577 

1.7675 

30 

31 

733 

.0950 

894 

.0042 

094 

.9196 

333 

.8405 

616 

.7663 

29 

32 

769 

.0934 

931 

.0028 

131 

.9183 

371 

.8392 

654 

.7651 

28 

33 

805 

.0918 

.49967 

2.0013 

168 

.9169 

409 

.8379 

693 

.7639 

27 

34 

840 

.0903 

.50004 

1.9999 

205 

.9155 

446 

.8367 

731 

.7627 

26 

35 

.47876 

2.0887 

.50040 

1.9984 

.52242 

1.9142 

.54484 

1.8354 

.56769 

1.7615 

25 

36 

912 

.0872 

076 

.9970 

279 

.9128 

522 

.8341 

808 

.7603 

24 

37 

948 

.0856 

113 

.9955 

316 

.9115 

560 

.8329 

846 

.7591 

23 

38 

.47984 

.0840 

149 

.9941 

353 

.9101 

597 

.8316 

885 

.7579 

22 

39 

.48019 

.0825 

185 

.9926 

390 

.9088 

635 

.8303 

923 

.7567 

21 

40 

.48055 

2.0809 

.50222 

1.9912 

.52427 

1.9074 

.54673 

1.8291 

.56962 

1.7556 

20 

41 

091 

.0794 

258 

.9897 

464 

.9061 

711 

.8278 

.57000 

.7544 

19 

42 

127 

.0778 

295 

.9883 

501 

.9047 

748 

.8265 

039 

.7532 

18 

43 

163 

.0763 

331 

.9868 

538 

.9034 

786 

.8253 

078 

.7520 

17 

44 

198 

.0748 

368 

.9854 

575 

.9020 

824 

.8240 

116 

.7508 

16 

45 

.48234 

2.0732 

.50404 

1.9840 

.52613 

1.9007 

.54862 

1.8228 

.57155 

1.7496 

15 

46 

270 

.0717 

441 

.9825 

650 

.8993 

900 

.8215 

193 

.7485 

14 

47 

306 

.0701 

477 

.9811 

687 

.8980 

938 

.8202 

232 

.7473 

13 

48 

342 

.0686 

514 

.9797 

724 

.8967 

.54975 

.8190 

271 

.7461 

12 

49 

378 

.€671 

550 

.9782 

761 

.8953 

.55013 

.8177 

309 

.7449 

11 

50 

.48414 

2.0655 

.50587 

1.9768 

.52798 

1.8940 

.55051 

1.8165 

.57348 

1.7437 

10 

51 

450 

.0640 

623 

.9754 

836 

.8927 

089 

.8152 

386 

.7426 

9 

52 

486 

.0625 

660 

.9740 

873 

.8913 

127 

.8140 

425 

.7414 

8 

53 

521 

.0609 

696 

.9725 

910 

.8900 

165 

.8127 

464 

.7402 

7 

54 

557 

.0594 

7.33 

.9711 

947 

.8887 

203 

.8115 

503 

.7391 

6 

55 

.48593 

2.0579 

.50769 

1.9697 

.52985 

1.8873 

.55241 

1.8103 

.57541 

1.7379 

5 

56 

629 

.0564 

806 

.9683 

.53022 

.8860 

279 

.8090 

580 

.7367 

4 

57 

665 

.0549 

843 

.9669 

059 

.8847 

317 

.8078 

619 

.7355 

3 

58 

701 

.0533 

879 

.9654 

096 

.8834 

355 

.8065 

657 

.7344 

2 

59 

737 

.0518 

916 

.9640 

134 

.8820 

393 

.8053 

696 

.7332 

1 

60 

.48773 

2.0503 

.50953 

1.9626 

.53171 

1.8807 

.55431 

1.8040 

.57735 

1.7321 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 



64° 

63 

o 

62 

o 

6r 

> 

60° 



[ 99 ] 















































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



30° 

31° 

32° 

33° 

CO 

o 



Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.50000 

.86603 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

60 

1 

025 

588 

529 

702 

.53017 

789 

488 

851 

943 

887 

59 

2 

050 

573 

554 

687 

041 

774 

513 

835 

968 

871 

58 

3 

076 

559 

579 

672 

066 

759 

537 

819 

.55992 

855 

57 

4 

101 

544 

604 

657 

091 

743 

561 

804 

.56016 

839 

56 

5 

.50126 

.86530 

.51628 

.85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.82822 

55 

6 

151 

515 

653 

627 

140 

712 

610 

772 

064 

806 

54 

7 

176 

501 

678 

612 

164 

697 

635 

756 

088 

790 

53 

8 

201 

486 

703 

597 

189 

681 

659 

740 

112 

773 

52 

9 

227 

471 

728 

582 

214 

666 

683 

724 

136 

757 

51 

10 

.50252 

.86457 

.51753 

.85567 

.53238 

.84650 

.54708 

.83708 

.56160 

.82741 

50 

11 

277 

442 

778 

551 

263 

635 

732 

692 

184 

724 

49 

12 

302 

427 

803 

536 

288 

619 

756 

676 

208 

708 

48 

13 

327 

413 

828 

521 

312 

604 

781 

660 

232 

692 

47 

14 

352 

398 

852 

506 

337 

588 

805 

645 

256 

675 

46 

15 

.50377 

.86384 

.51877 

.85491 

.53361 

.84573 

.54829 

.83629 

.56280 

.82659 

45 

16 

403 

369 

902 

476 

386 

557 

854 

613 

305 

643 

44 

17 

428 

354 

927 

461 

411 

542 

878 

597 

329 

626 

43 

18 

453 

340 

952 

446 

435 

526 

902 

581 

353 

610 

42 

19 

478 

325 

.51977 

431 

460 

511 

927 

565 

377 

593 

41 

20 

.50503 

.86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

528 

295 

026 

401 

509 

480 

975 

533 

425 

561 

39 

22 

553 

281 

051 

385 

534 

464 

.54999 

517 

449 

544 

38 

23 

578 

266 

076 

370 

558 

448 

.55024 

501 

473 

528 

37 

24 

603 

251 

101 

355 

583 

433 

048 

485 

497 

511 

36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495 

35 

26 

654 

222 

151 

325 

632 

402 

097 

453 

545 

478 

34 

27 

679 

207 

175 

310 

656 

386 

121 

437 

569 

462 

33 

28 

704 

192 

200 

294 

681 

370 

145 

421 

593 

446 

32 

29 

729 

178 

225 

279 

705 

355 

169 

405 

617 

429 

31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

779 

148 

275 

249 

754 

324 

218 

373 

665 

396 

29 

32 

804 

133 

299 

234 

779 

308 

242 

356 

689 

380 

28 

33 

829 

119 

324 

218 

804 

292 

266 

340 

713 

363 

27 

34 

854 

104 

349 

203 

828 

277 

291 

324 

736 

347 

26 

35 

.50879 

.86089 

.52374 

.85188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

904 

074 

399 

173 

877 

245 

339 

292 

784 

314 

24 

37 

929 

059 

423 

157 

902 

230 

363 

276 

808 

297 

23 

38 

954 

045 

448 

142 

926 

214 

388 

260 

832 

281 

22 

39 

.50979 

030 

473 

127 

951 

198 

412 

244 

856 

264 

21 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

029 

.86000 

522 

096 

.54000 

167 

460 

212 

904 

231 

19 

42 

054 

.85985 

547 

081 

024 

151 

484 

195 

928 

214 

18 

43 

079 

970 

572 

066 

049 

135 

509 

179 

952 

198 

17 

44 

104 

956 

597 

051 

073 

120 

533 

163 

.56976 

181 

16 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.83147 

.57000 

.82165 

15 

46 

154 

926 

646 

020 

122 

088 

581 

131 

024 

148 

14 

47 

179 

911 

671 

.85005 

146 

072 

605 

115 

047 

132 

13 

48 

204 

896 

696 

.84989 

171 

057 

630 

098 

071 

115 

12 

49 

229 

881 

720 

974 

195 

041 

654 

082 

095 

098 

11 

50 

.51254 

.85866 

.52745 

.84959 

.54220 

.84025 

.55678 

.83066 

.57119 

.82082 

10 

51 

279 

851 

770 

943 

244 

.84009 

702 

050 

143 

065 

9 

52 

304 

836 

794 

928 

269 

.83994 

726 

034 

167 

048 

8 

53 

329 

821 

819 

913 

293 

978 

750 

017 

191 

032 

7 

54 

354 

806 

844 

897 

317 

962 

775 

.83001 

215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

404 

777 

893 

866 

366 

930 

823 

969 

262 

982 

4 

57 

429 

762 

918 

851 

391 

915 

847 

953 

286 

965 

3 

58 

454 

747 

943 

836 

415 

899 

871 

936 

310 

949 

2 

59 

479 

732 

967 

820 

440 

883 

895 

920 

334 

932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


59° 

o 

co 

lO 

57° 

56 

o 

55 

o 



[ 100 ] 

















































































































XI. FIVE PLACE VALUES: TANGENT AND COTANGENT 



i 

50° 

1 

CO 

O 

32° 

33° 

34° 


1 / 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

n 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

.57735 

774 

813 

851 

890 

.57929 

.57968 

.58007 

046 

085 

.58124 

1.7321 

.7309 

.7297 

.7286 

.7274 

1.7262 

.7251 

.7239 

.7228 

.7216 

1.7205 

.60086 

126 

165 

205 

245 

.60284 

324 

364 

403 

443 

.60483 

1.6643 

.6632 

.6621 

.6610 

.6599 

1.6588 

.6577 

.6566 

.6555 

.6545 

1.6534 

.62487 

527 

568 

608 

649 

.62689 

730 

770 

811 

852 

.62892 

933 

.62973 

.63014 

055 

.63095 

136 

177 

217 

258 

.63299 

1.6003 

.5993 

.5983 

.5972 

.5962 

1.5952 

.5941 

.5931 

.5921 

.5911 

1.5900 

.64941 

.64982 

.65024 

065 

106 

.65148 

189 

231 

272 

314 

.65355 

1.5399 

.5389 

.5379 

.5369 

.5359 

1.5350 

.5340 

.5330 

.5320 

.5311 

1.5301 

.67451 

493 

536 

578 

620 

.67663 

705 

748 

790 

832 

.67875 

1.4826 

.4816 

.4807 

.4798 

.4788 

1.4779 

.4770 

.4761 

.4751 

.4742 

1.4733 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

162 

201 

240 

279 

.58318 

357 

396 

435 

474 

.58513 

.7193 

.7182 

.7170 

.7159 

1.7147 

.7136 

.7124 

.7113 

.7102 

1.7090 

522 

562 

602 

642 

.60681 

721 

761 

801 

841 

.60881 

.6523 

.6512 

.6501 

.6490 

1.6479 

.6469 

.6458 

.6447 

.6436 

1.6426 

.5890 

.5880 

.5869 

.5859 

1.5849 

.5839 

.5829 

.5818 

.5808 

1.5798 

397 

438 

480 

521 

.65563 

604 

646 

688 

729 

.65771 

.5291 

.5282 

.5272 

.5262 

1.5253 

.5243 

.5233 

.5224 

.5214 

1.5204 

917 

.67960 

.68002 

045 

.68088 

130 

173 

215 

258 

.68301 

.4724 

.4715 

.4705 

.4696 

1.4687 

.4678 

.4669 

.4659 

.4650 

1.4641 

552 

591 

631 

670 

.58709 

748 

787 

826 

865 

.58905 

.7079 

.7067 

.7056 

.7045 

1.7033 

.7022 

.7011 

.6999 

.6988 

1.6977 

921 

.60960 

.61000 

040 

.61080 

120 

160 

200 

240 

.61280 

.6415 

.6404 

.6393 

.6383 

1.6372 

.6361 

.6351 

.6340 

.6329 

1.6319 

340 

380 

421 

462 

.63503 

544 

584 

625 

666 

.63707 

.5788 

.5778 

.5768 

.5757 

1.5747 

.5737 

.5727 

.5717 

.5707 

1.5697 

813 

854 

896 

938 

.65980 

.66021 

063 

105 

147 

.66189 

.5195 

.5185 

.5175 

.5166 

1.5156 

.5147 

.5137 

.5127 

.5118 

1.5108 

343 

386 

429 

471 

.68514 

557 

600 

642 

685 

.68728 

.4632 

.4623 

.4614 

.4605 

1.4596 

.4586 

.4577 

.4568 

.4559 

1.4550 

31 

944 

.6965 

320 

.6308 

748 

.5687 

230 

.5099 

771 

4541 

2Q 1 

| 32 

.58983 

.6954 

360 

.6297 

789 

.5677 

272 

.5089 

814 

.4532 

28 

| 33 

.59022 

.6943 

400 

.6287 

830 

.5667 

314 

.5080 

857 

.4523 

27 

1 34 

061 

.6932 

440 

.6276 

871 

.5657 

356 

.5070 

900 

.4514 

26 

I 35 

.59101 

1.6920 

.61480 

1.6265 

.63912 

1.5647 

.66398 

1.5061 

.68942 

1.4505 

25 

36 

140 

.6909 

520 

.6255 

953 

.5637 

440 

.5051 

.68985 

.4496 

24 I 

| 37 

179 

.6898 

561 

.6244 

.63994 

.5627 

482 

.5042 

.69028 

.4487 

23 I 

1 38 

218 

.6887 

601 

.6234 

.64035 

.5617 

524 

.5032 

071 

.4478 

22 

1 39 

258 

.6875 

641 

.6223 

076 

.5607 

566 

.5023 

114 

.4469 

21 

40 

.59297 

1.6864 

.61681 

1.6212 

.64117 

1.5597 

.66608 

1.5013 

.69157 

1.4460 

20 

41 

336 

.6853 

721 

.6202 

158 

.5587 

650 

.5004 

200 

.4451 

19 I 

42 

376 

.6842 

761 

.6191 

199 

.5577 

692 

.4994 

243 

.4442 

18 

1 43 

415 

.6831 

801 

.6181 

240 

.5567 

734 

.4985 

286 

.4433 

17 

| 44 

454 

.6820 

842 

.6170 

281 

.5557 

776 

.4975 

329 

.4424 

16 

45 

.59494 

1.6808 

.61882 

1.6160 

.64322 

1.5547 

.66818 

1.4966 

.69372 

1.4415 

15| 

46 

533 

.6797 

922 

.6149 

363 

.5537 

860 

.4957 

416 

.4406 

14 

1 47 

573 

.6786 

.61962 

.6139 

404 

.5527 

902 

.4947 

459 

.4397 

13 

| 48 

612 

.6775 

.62003 

.6128 

446 

.5517 

944 

.4938 

502 

.4388 

12 

| 49 

651 

.6764 

043 

.6118 

487 

.5507 

.66986 

.4928 

545 

.4379 

11 

50 

.59691 

1.6753 

.62083 

1.6107 

.64528 

1.5497 

.67028 

1.4919 

.69588 

1.4370 

10 1 

51 

730 

.6742 

124 

.6097 

569 

.5487 

071 

.4910 

631 

.4361 

9 

| 52 

770 

.6731 

164 

.6087 

610 

.5477 

113 

.4900 

675 

.4352 

8] 

| 53 

809 

.6720 

204 

.6076 

652 

.5468 

155 

.4891 

718 

.4344 

7 

54 

849 

.6709 

245 

.6066 

693 

.5458 

197 

.4882 

761 

.4335 

6 

55 

.59888 

1.6698 

.62285 

1.6055 

.64734 

1.5448 

.67239 

1.4872 

.69804 

1.4326 

5| 

56 

928 

.6687 

325 

.6045 

775 

.5438 

282 

.4863 

847 

.4317 

4 

57 

.59967 

.6676 

366 

.6034 

817 

.5428 

324 

.4854 

891 

.4308 

31 

58 

.60007 

.6665 

406 

.6024 

858 

.5418 

366 

.4844 

934 

.4299 

21 

59 

046 

.6654 

446 

.6014 

899 

.5408 

409 

.4835 

.69977 

.4290 

n 

60 

.60086 

1.6643 

.62487 

1.6003 

.64941 

1.5399 

.67451 

1.4826 

.70021 

1.4281 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 1 


59 

D 

58° 

57° 

56° 

55° 

<- 


[ 101 ] 















































































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



35° 

36° 

37° 

CO 

00 

o 

39° 



Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.57358 

.81915 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

60 

1 

381 

899 

802 

885 

205 

846 

589 

783 

955 

696 

59 

2 

405 

882 

826 

867 

228 

829 

612 

765 

.62977 

678 

58 

3 

429 

865 

849 

850 

251 

811 

635 

747 

.63000 

660 

57 

4 

453 

848 

873 

833 

274 

793 

658 

729 

022 

641 

56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681 

.78711 

.63045 

.77623 

55 

6 

501 

815 

920 

799 

321 

758 

704 

694 

068 

605 

54 

7 

524 

798 

943 

782 

344 

741 

726 

676 

090 

586 

53 

8 

548 

782 

967 

765 

367 

723 

749 

658 

113 

568 

52 

9 

572 

765 

.58990 

748 

390 

706 

772 

640 

135 

550 

51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531 

50 

11 

619 

731 

037 

713 

437 

671 

818 

604 

180 

513 

49 

12 

643 

714 

061 

696 

460 

653 

841 

586 

203 

494 

48 

13 

667 

698 

084 

679 

483 

635 

864 

568 

225 

476 

47 

14 

691 

681 

108 

662 

506 

618 

887 

550 

248 

458 

46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909 

.78532 

.63271 

.77439 

45 

16 

738 

647 

154 

627 

553 

583 

932 

514 

293 

421 

44 

17 

762 

631 

178 

610 

576 

565 

955 

496 

316 

402 

43 

18 

786 

614 

201 

593 

599 

547 

.61978 

478 

338 

384 

42 

19 

810 

597 

225 

576 

622 

530 

.62001 

460 

361 

366 

41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

857 

563 

272 

541 

668 

494 

046 

424 

406 

329 

39 

22 

881 

546 

295 

524 

691 

477 

069 

405 

428 

310 

38 

23 

904 

530 

318 

507 

714 

459 

092 

387 

451 

292 

37 

24 

928 

513 

342 

489 

738 

441 

115 

369 

473 

273 

36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255 

35 

26 

976 

479 

389 

455 

784 

406 

160 

333 

518 

236 

34 

27 

.57999 

462 

412 

438 

807 

388 

183 

315 

540 

218 

33 

28 

.58023 

445 

436 

420 

830 

371 

206 

297 

563 

199 

32 

29 

047 

428 

459 

403 

853 

353 

229 

279 

585 

181 

31 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

094 

395 

506 

368 

899 

318 

274 

243 

630 

144 

29 

32 

118 

378 

529 

351 

922 

300 

297 

225 

653 

125 

28 

33 

141 

361 

552 

334 

945 

282 

320 

206 

675 

107 

27 

34 

165 

344 

576 

316 

968 

264 

342 

188 

698 

088 

26 

35 

.58189 

.81327 

.59599 

.80299 

.60991 

.79247 

.62365 

.78170 

.63720 

.77070 

25 

36 

212 

' 310 

622 

282 

.61015 

229 

388 

152 

742 

051 

24 

37 

236 

293 

646 

264 

038 

211 

411 

134 

765 

033 

23 

38 

260 

276 

669 

247 

061 

193 

433 

116 

787 

.77014 

22 

39 

283 

259 

693 

230 

084 

176 

456 

098 

810 

.76996 

21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158 

.62479 

.78079 

.63832 

.76977 

20 

41 

330 

225 

739 

195 

130 

140 

502 

061 

854 

959 

19 

42 

354 

208 

763 

178 

153 

122 

524 

043 

877 

940 

18 

43 

378 

191 

786 

160 

176 

105 

547 

025 

899 

921 

17 

44 

401 

174 

809 

143 

199 

087 

570 

.78007 

922 

903 

16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.77988 

.63944 

.76884 

15 

46 

449 

140 

856 

108 

245 

051 

615 

970 

966 

866 

14 

47 

472 

123 

879 

091 

268 

033 

638 

952 

.63989 

847 

13 

48 

496 

106 

902 

073 

291 

.79016 

660 

934 

.64011 

828 

12 

49 

519 

089 

926 

056 

314 

.78998 

683 

916 

033 

810 

11 

50 

.58543 

.81072 

.59949 

.80038 

.61337 

.78980 

.62706 

.77897 

.64056 

.76791 

10 

51 

567 

055 

972 

021 

360 

962 

728 

879 

078 

772 

9 

52 

590 

038 

.59995 

.80003 

383 

944 

751 

861 

100 

754 

8 

53 

614 

021 

.60019 

.79986 

406 

926 

774 

843 

123 

735 

7 

54 

637 

.81004 

042 

968 

429 

908 

796 

824 

145 

717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

56 

684 

970 

089 

934 

474 

873 

842 

788 

190 

679 

4 

57 

708 

953 

112 

916 

497 

855 

864 

769 

212 

661 

3 

58 

731 

936 

135 

899 

520 

837 

887 

751 

234 

642 

2 

59 

755 

919 

158 

881 

543 

819 

909 

733 

256 

623 

1 

60 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715 

.64279 

.76604 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

f 


54° 

53° 

52° 

51° 

50° 



[ 102 ] 





































































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 


-> 

35° 

36° 

37° 

38° 

39° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

.70021 

064 

107 

151 

194 

.70238 

281 

325 

368 

412 

.70455 

1.4281 

.4273 

.4264 

.4255 

.4246 

1.4237 

.4229 

.4220 

.4211 

.4202 

1.4193 

.72654 

699 

743 

788 

832 

.72877 

921 

.72966 

.73010 

055 

.73100 

1.3764 

.3755 

.3747 

.3739 

.3730 

1.3722 

.3713 

.3705 

.3697 

.3688 

1.3680 

.75355 

401 

447 

492 

538 

.75584 

629 

675 

721 

767 

.75812 

1.3270 

.3262 

.3254 

.3246 

.3238 

1.3230 

.3222 

.3214 

.3206 

.3198 

1.3190 

.78129 

175 

222 

269 

316 

.78363 

410 

457 

504 

551 

.78598 

1.2799 

.2792 

.2784 

.2776 

.2769 

1.2761 

.2753 

.2746 

.2738 

.2731 

1.2723 

.80978 

.81027 

075 

123 

171 

.81220 

268 

316 

364 

413 

.81461 

1.2349 

.2342 

.2334 

.2327 

.2320 

1.2312 

.2305 

.2298 

.2290 

.2283 

1.2276 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

499 

542 

586 

629 

.70673 

717 

760 

804 

848 

.70891 

.4185 

.4176 

.4167 

.4158 

1.4150 

.4141 

.4132 

.4124 

.4115 

1.4106 

144 

189 

234 

278 

.73323 

368 

413 

457 

502 

.73547 

.3672 

.3663 

.3655 

.3647 

1.3638 

.3630 

.3622 

.3613 

.3605 

1.3597 

858 

904 

950 

.75996 

.76042 

088 

134 

180 

226 

.76272 

.3182 

.3175 

.3167 

.3159 

1.3151 

.3143 

.3135 

.3127 

.3119 

1.3111 

645 

692 

739 

786 

.78834 

881 

928 

.78975 

.79022 

.79070 

.2715 

.2708 

.2700 

.2693 

1.2685 

.2677 

.2670 

.2662 

.2655 

1.2647 

510 

558 

606 

655 

.81703 

752 

800 

849 

898 

.81946 

.2268 

.2261 

.2254 

.2247 

1.2239 

.2232 

.2225 

.2218 

.2210 

1.2203 

49 

48 

47 

46 

45 

44 

43 

42 

41 

40 

39 

38 

37 

36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

0 

935 

.70979 

.71023 

066 

.71110 

154 

198 

242 

285 

.71329 

.4097 

.4089 

.4080 

.4071 

1.4063 

.4054 

.4045 

.4037 

.4028 

1.4019 

592 

637 

681 

726 

.73771 

816 

861 

906 

951 

.73996 

.3588 

.3580 

.3572 

.3564 

1.3555 

.3547 

.3539 

.3531 

.3522 

1.3514 

318 

364 

410 

456 

.76502 

548 

594 

640 

686 

.76733 

.3103 

.3095 

.3087 

.3079 

1.3072 

.3064 

.3056 

.3048 

.3040 

1.3032 

117 

164 

212 

259 

.79306 

354 

401 

449 

496 

.79544 

.2640 

.2632 

.2624 

.2617 

1.2609 

.2602 

.2594 

.2587 

.2579 

1.2572 

.81995 

.82044 

092 

141 

.82190 

238 

287 

336 

385 

.82434 

.2196 

.2189 

.2181 

.2174 

1.2167 

.2160 

.2153 

.2145 

.2138 

1.2131 

373 

417 

461 

505 

.71549 

593 

637 

681 

725 

.71769 

.4011 

.4002 

.3994 

.3985 

1.3976 

.3968 

.3959 

.3951 

.3942 

1.3934 

.74041 

086 

131 

176 

.74221 

267 

312 

357 

402 

.74447 

.3506 

.3498 

.3490 

.3481 

1.3473 

.3465 

.3457 

.3449 

.3440 

1.3432 

779 

825 

871 

918 

.76964 

.77010 

057 

103 

149 

.77196 

.3024 

.3017 

.3009 

.3001 

1.2993 

.2985 

.2977 

.2970 

.2962 

1.2954 

591 

639 

686 

734 

.79781 

829 

877 

924 

.79972 

.80020 

.2564 

.2557 

.2549 

.2542 

1.2534 

.2527 

.2519 

.2512 

.2504 

1.2497 

483 

531 

580 

629 

.82678 

727 

776 

825 

874 

.82923 

.2124 

.2117 

.2109 

.2102 

1.2095 

.2088 

.2081 

.2074 

.2066 

1.2059 

813 

857 

901 

946 

.71990 

.72034 

078 

122 

167 

.72211 

.3925 

.3916 

.3908 

.3899 

1.3891 

.3882 

.3874 

.3865 

.3857 

1.3848 

492 

538 

583 

628 

.74674 

719 

764 

810 

855 

.74900 

.3424 

.3416 

.3408 

.3400 

1.3392 

.3384 

.3375 

.3367 

.3359 

1.3351 

242 

289 

335 

382 

.77428 

475 

521 

568 

615 

.77661 

.2946 

.2938 

.2931 

.2923 

1.2915 

.2907 

.2900 

.2892 

.2884 

1.2876 

067 

115 

163 

211 

.80258 

306 

354 

402 

450 

.80498 

.2489 

.2482 

.2475 

.2467 

1.2460 

.2452 

.2445 

.2437 

.2430 

1.2423 

.82972 

.83022 

071 

120 

.83169 

218 

268 

317 

366 

.83415 

.2052 

.2045 

.2038 

.2031 

1.2024 

.2017 

.2009 

.2002 

.1995 

1.1988 

255 

299 

344 

388 

.72432 

477 

521 

565 

610 

.72654 

.3840 

.3831 

.3823 

.3814 

1.3806 

.3798 

.3789 

.3781 

.3772 

1.3764 

946 

.74991 

.75037 

082 

.75128 

173 

219 

264 

310 

.75355 

.3343 

.3335 

.3327 

.3319 

1.3311 

.3303 

.3295 

.3287 

.3278 

1.3270 

708 

754 

801 

848 

.77895 

941 

.77988 

.78035 

082 

.78129 

.2869 

.2861 

.2853 

.2846 

1.2838 

.2830 

.2822 

.2815 

.2807 

1.2799 

546 

594 

642 

690 

.80738 

786 

834 

882 

930 

.80978 

.2415 

.2408 

.2401 

.2393 

1.2386 

.2378 

.2371 

.2364 

.2356 

1.2349 

465 

514 

564 

613 

.83662 

712 

761 

811 

860 

.83910 

.1981 

.1974 

.1967 

.1960 

1.1953 

.1946 

.1939 

.1932 

.1925 

1.1918 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 


54° 

53° 

52° 

51° 

50° 

<- 


C 103 ] 




























































































































XI. FIVE-PLACE VALUES: SINE AND COSINE 



o 

o 

o 

42° 

43° 

44° 


/ 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 


0 

.64279 

.76604 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

60 

1 

301 

586 

628 

452 

935 

295 

221 

116 

487 

914 

59 

2 

323 

567 

650 

433 

956 

276 

242 

096 

508 

894 

58 

3 

346 

548 

672 

414 

978 

256 

264 

076 

529 

873 

57 

4 

368 

530 

694 

395 

.66999 

237 

285 

056 

549 

853 

56 

5 

.64390 

.76511 

.65716 

.75375 

.67021 

.74217 

.68306 

.73036 

.69570 

.71833 

55 

6 

412 

492 

738 

356 

043 

198 

327 

.73016 

591 

813 

54 

7 

435 

473 

759 

337 

064 

178 

349 

.72996 

612 

792 

53 

8 

457 

455 

781 

318 

086 

159 

370 

976 

633 

772 

52 

9 

479 

436 

803 

299 

107 

139 

391 

957 

654 

752 

51 

10 

.64501 

.76417 

.65825 

.75280 

.67129 

.74120 

.68412 

.72937 

.69675 

.71732 

50 

11 

524 

398 

847 

261 

151 

100 

434 

917 

696 

711 

49 

12 

546 

380 

869 

241 

172 

080 

455 

897 

717 

691 

48 

13 

568 

361 

891 

222 

194 

061 

476 

877 

737 

671 

47 

14 

590 

342 

913 

203 

215 

041 

497 

857 

758 

650 

46 

15 

.64612 

.76323 

.65935 

.75184 

.67237 

.74022 

.68518 

.72837 

.69779 

.71630 

45 

16 

635 

304 

956 

165 

258 

.74002 

539 

817 

800 

610 

44 

17 

657 

286 

.65978 

146 

280 

.73983 

561 

797 

821 

590 

43 

18 

679 

267 

.66000 

126 

301 

963 

582 

777 

842 

569 

42 

19 

701 

248 

022 

107 

323 

944 

603 

757 

862 

549 

41 

20 

.64723 

.76229 

.66044 

.75088 

.67344 

.73924 

.68624 

.72737 

.69883 

.71529 

40 

21 

746 

210 

066 

069 

366 

904 

645 

717 

904 

508 

39 

22 

768 

192 

088 

050 

387 

885 

666 

697 

925 

488 

38 

23 

790 

173 

109 

030 

409 

865 

688 

677 

946 

468 

37 

24 

812 

154 

131 

.75011 

430 

846 

709 

657 

966 

447 

36 

25 

.64834 

.76135 

.66153 

.74992 

.67452 

.73826 

.68730 

.72637 

.69987 

.71427 

35 

26 

856 

116 

175 

973 

473 

806 

751 

617 

.70008 

407 

34 

27 

878 

097 

197 

953 

495 

787 

772 

597 

029 

386 

33 

28 

901 

078 

218 

934 

516 

767 

793 

577 

049 

366 

32 

29 

923 

059 

240 

915 

538 

747 

814 

557 

070 

345 

31 

30 

.64945 

.76041 

.66262 

.74896 

.67559 

.73728 

.68835 

.72537 

.70091 

.71325 

30 

31 

967 

022 

284 

876 

580 

708 

857 

517 

112 

305 

29 

32 

.64989 

.76003 

306 

857 

602 

688 

878 

497 

132 

284 

28 

33 

.65011 

.75984 

327 

838 

623 

669 

899 

477 

153 

264 

27 

34 

033 

965 

349 

818 

645 

649 

920 

457 

174 

243 

26 

35 

.65055 

.75946 

.66371 

.74799 

.67666 

.73629 

.68941 

.72437 

.70195 

.71223 

25 

36 

077 

927 

393 

780 

688 

610 

962 

417 

215 

203 

24 

37 

100 

908 

414 

760 

709 

590 

.68983 

397 

236 

182 

23 

38 

122 

889 

436 

741 

730 

570 

.69004 

377 

257 

162 

22 

39 

144 

870 

458 

722 

752 

551 

025 

357 

277 

141 

21 

40 

.65166 

.75851 

.66480 

.74703 

.67773 

.73531 

.69046 

.72337 

.70298 

.71121 

20 

41 

188 

832 

501 

683 

795 

511 

067 

317 

319 

100 

19 

42 

210 

813 

523 

664 

816 

491 

088 

297 

339 

080 

18 

43 

232 

794 

545 

644 

837 

472 

109 

277 

360 

059 

17 

44 

254 

775 

566 

625 

859 

452 

130 

257 

381 

039 

16 

45 

.65276 

.75756 

.66588 

.74606 

.67880 

.73432 

.69151 

.72236 

.70401 

.71019 

15 

46 

298 

738 

610 

586 

901 

413 

172 

216 

422 

.70998 

14 

47 

320 

719 

632 

567 

923 

393 

193 

196 

443 

978 

13 

48 

342 

700 

653 

548 

944 

373 

214 

176 

463 

957 

12 

49 

364 

680 

675 

528 

965 

353 

235 

156 

484 

937 

11 

50 

.65386 

.75661 

.66697 

.74509 

.67987 

.73333 

.69256 

.72136 

.70505 

.70916 

10 

51 

408 

642 

718 

489 

.68008 

314 

277 

116 

525 

896 

9 

52 

430 

623 

740 

470 

029 

294 

298 

095 

546 

875 

8 

53 

452 

604 

762 

451 

051 

274 

319 

075 

567 

855 

7 

54 

474 

585 

783 

431 

072 

254 

340 

055 

587 

834 

6 

55 

.65496 

.75566 

.66805 

.74412 

.68093 

.73234 

.69361 

.72035 

.70608 

.70813 

5 

56 

518 

547 

827 

392 

115 

215 

382 

.72015 

628 

793 

4 

57 

540 

528 

848 

373 

136 

195 

403 

.71995 

649 

772 

3 

58 

562 

509 

870 

353 

157 

175 

424 

974 

670 

752 

2 

59 

584 

490 

891 

334 

179 

155 

445 

954 

690 

731 

1 

60 

.65606 

.75471 

.66913 

.74314 

.68200 

.73135 

.69466 

.71934 

.70711 

.70711 

0 


Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

Cos 

Sin 

/ 


49° 

o 

00 

47° 

46° 

45° 



[ 104 ] 












































































































XI. FIVE-PLACE VALUES: TANGENT AND COTANGENT 



o 

0 

41° 

42° 

43° 

44° 


/ 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 


0 

.83910 

1.1918 

.86929 

1.1504 

.90040 

1.1106 

.93252 

1.0724 

.96569 

1.0355 

60 

1 

.83960 

.1910 

.86980 

.1497 

093 

.1100 

306 

.0717 

625 

.0349 

59 

i 2 

.84009 

.1903 

.87031 

.1490 

146 

.1093 

360 

.0711 

681 

.0343 

58 

3 

059 

.1896 

082 

.1483 

199 

.1087 

415 

.0705 

738 

.0337 

57 

4 

108 

.1889 

133 

.1477 

251 

.1080 

469 

.0699 

794 

.0331 

56 

5 

.84158 

1.1882 

.87184 

1.1470 

.90304 

1.1074 

.93524 

1.0692 

.96850 

1.0325 

55 

6 

208 

.1875 

236 

.1463 

357 

.1067 

578 

.0686 

907 

.0319 

54 

'! 7 

258 

.1868 

287 

.1456 

410 

.1061 

633 

.0680 

.96963 

.0313 

53 

8 

307 

.1861 

338 

.1450 

463 

.1054 

688 

.0674 

.97020 

.0307 

52 

9 

357 

.1854 

389 

.1443 

516 

.1048 

742 

.0668 

076 

.0301 

51 

10 

.84407 

1.1847 

.87441 

1.1436 

.90569 

1.1041 

.93797 

1.0661 

.97133 

1.0295 

50 

11 

457 

.1840 

492 

.1430 

621 

.1035 

852 

.0655 

189 

.0289 

49 

12 

507 

.1833 

543 

.1423 

674 

.1028 

906 

.0649 

246 

.0283 

48 

13 

556 

.1826 

595 

.1416 

727 

.1022 

.93961 

.0643 

302 

.0277 

47 

14 

606 

.1819 

646 

.1410 

781 

.1016 

.94016 

.0637 

359 

.0271 

46 

15 

.84656 

1.1812 

.87698 

1.1403 

.90834 

1.1009 

.94071 

1.0630 

.97416 

1.0265 

45 

16 

706 

.1806 

749 

.1396 

887 

.1003 

125 

.0624 

472 

.0259 

44 

17 

756 

.1799 

801 

.1389 

940 

.0996 

180 

.0618 

529 

.0253 

43 

18 

806 

.1792 

852 

.1383 

.90993 

.0990 

235 

.0612 

586 

.0247 

42 

19 

856 

.1785 

904 

.1376 

.91046 

.0983 

290 

.0606 

643 

.0241 

41 

20 

.84906 

1.1778 

.87955 

1.1369 

.91099 

1.0977 

.94345 

1.0599 

.97700 

1.0235 

40 

21 

.84956 

.1771 

.88007 

.1363 

153 

.0971 

400 

.0593 

756 

.0230 

39 

22 

.85006 

.1764 

059 

.1356 

206 

.0964 

455 

.0587 

813 

.0224 

38 

23 

057 

.1757 

110 

.1349 

259 

.0958 

510 

.0581 

870 

.0218 

37 

24 

107 

.1750 

162 

.1343 

313 

.0951 

565 

.0575 

927 

.0212 

36 

25 

.85157 

1.1743 

.88214 

1.1336 

.91366 

1.0945 

.94620 

1.0569 

.97984 

1.0206 

35 

26 

207 

.1736 

265 

.1329 

419 

.0939 

676 

.0562 

.98041 

.0200 

34 

27 

257 

.1729 

317 

.1323 

473 

.0932 

731 

.0556 

098 

.0194 

33 

28 

308 

.1722 

369 

.1316 

526 

.0926 

786 

.0550 

155 

.0188 

32 

29 

358 

.1715 

421 

.1310 

580 

.0919 

841 

.0544 

213 

.0182 

31 

30 

.85408 

1.1708 

.88473 

1.1303 

.91633 

1.0913 

.94896 

1.0538 

.98270 

1.0176 

30 

31 

458 

.1702 

524 

.1296 

687 

.0907 

.94952 

.0532 

327 

.0170 

29 

32 

509 

.1695 

576 

.1290 

740 

.0900 

.95007 

.0526 

384 

.0164 

28 

33 

559 

.1688 

628 

.1283 

794 

.0894 

062 

.0519 

441 

.0158 

27 

34 

609 

.1681 

680 

.1276 

847 

.0888 

118 

.0513 

499 

.0152 

26 

35 

.85660 

1.1674 

.88732 

1.1270 

.91901 

1.0881 

.95173 

1.0507 

.98556 

1.0147 

25 

36 

710 

.1667 

784 

.1263 

.91955 

.0875 

229 

.0501 

613 

.0141 

24 

37 

761 

.1660 

836 

.1257 

.92008 

.0869 

284 

.0495 

671 

.0135 

23 

38 

811 

.1653 

888 

.1250 

062 

.0862 

340 

.0489 

728 

.0129 

22 

39 

862 

.1647 

940 

.1243 

116 

.0856 

395 

.0483 

786 

.0123 

21 

40 

.85912 

1.1640 

.88992 

1.1237 

.92170 

1.0850 

.95451 

1.0477 

.98843 

1.0117 

20 

41 

.85963 

.1633 

.89045 

.1230 

224 

.0843 

506 

.0470 

901 

.0111 

19 

42 

.86014 

.1626 

097 

.1224 

277 

.0837 

562 

.0464 

.98958 

.0105 

18 

43 

064 

.1619 

149 

.1217 

331 

.0831 

618 

.0458 

.99016 

.0099 

17 

44 

115 

.1612 

201 

.1211 

385 

.0824 

673 

.0452 

073 

.0094 

16 

45 

.86166 

1.1606 

.89253 

1.1204 

.92439 

1.0818 

.95729 

1.0446 

.99131 

1.0088 

15 

46 

216 

.1599 

306 

.1197 

493 

.0812 

785 

.0440 

189 

.0082 

14 

47 

267 

.1592 

358 

.1191 

547 

.0805 

841 

.0434 

247 

.0076 

13 

48 

318 

.1585 

410 

.1184 

601 

.0799 

897 

.0428 

304 

.0070 

12 

49 

368 

.1578 

463 

.1178 

655 

.0793 

.95952 

.0422 

362 

.0064 

11 

50 

.86419 

1.1571 

.89515 

1.1171 

.92709 

1.0786 

.96008 

1.0416 

.99420 

1.0058 

10 

51 

470 

.1565 

567 

.1165 

763 

.0780 

064 

.0410 

478 

.0052 

9 

52 

521 

.1558 

620 

.1158 

817 

.0774 

120 

.0404 

536 

.0047 

8 

53 

572 

.1551 

672 

.1152 

872 

.0768 

176 

.0398 

594 

.0041 

7 

54 

623 

.1544 

725 

.1145 

926 

.0761 

232 

.0392 

652 

.0035 

6 

55 

.86674 

1.1538 

.89777 

1.1139 

.92980 

1.0755 

.96288 

1.0385 

.99710 

1.0029 

5 

56 

725 

.1531 

830 

.1132 

.93034 

.0749 

344 

.0379 

768 

.0023 

4 

57 

776 

.1524 

883 

.1126 

088 

.0742 

400 

.0373 

826 

.0017 

3 

58 

827 

.1517 

935 

.1119 

143 

.0736 

457 

.0367 

884 

.0012 

2 

59 

878 

.1510 

.89988 

.1113 

197 

.0730 

513 

.0361 

.99942 

.0006 

1 

60 

.86929 

1.1504 

.90040 

1.1106 

.93252 

1.0724 

.96569 

1.0355 

1.0000 

1.0000 

0 


Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

Cot 

Tan 

/ 


49° 

00 

o 

47 

o 

46 

o 

45 

o 



[ 105 ] 
































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



0 

O 

1 

o 

2 

o 

3 

o 

4 

_o 


/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.0000 


1.0002 

57.299 

1.0006 

28.654 

1.0014 

19.107 

1.0024 

14.336 

60 

1 

.0000 

3437.7 

.0002 

56.359 

.0006 

.417 

.0014 

19.002 

.0025 

.276 

59 

2 

.0000 

1718.9 

.0002 

55.451 

.0006 

28.184 

.0014 

18.898 

.0025 

.217 

58 

3 

.0000 

1145.9 

.0002 

54.570 

.0006 

27.955 

.0014 

.794 

.0025 

.159 

57 

4 

.0000 

859.44 

.0002 

53.718 

.0007 

.730 

.0014 

.692 

.0025 

.101 

56 

5 

1.0000 

687.55 

1.0002 

52.892 

1.0007 

27.508 

1.0014 

18.591 

1.0025 

14.044 

55 

6 

.0000 

572.96 

.0002 

52.090 

.0007 

.290 

.0015 

.492 

.0026 

13.987 

54 

7 

.0000 

491.11 

.0002 

51.313 

.0007 

27.075 

.0015 

.393 

.0026 

.930 

53 

8 

.0000 

429.72 

.0002 

50.558 

.0007 

26.864 

.0015 

.295 

.0026 

.874 

52 

9 

.0000 

381.97 

.0002 

49.826 

.0007 

.655 

.0015 

.198 

.0026 

.818 

51 

10 

1.0000 

343.78 

1.0002 

49.114 

1.0007 

26.451 

1.0015 

18.103 

1.0027 

13.763 

50 

11 

.0000 

312.52 

.0002 

48.422 

.0007 

.249 

.0015 

18.008 

.0027 

.708 

49 

12 

.0000 

286.48 

.0002 

47.750 

.0007 

26.050 

.0016 

17.914 

.0027 

.654 

48 

13 

.0000 

264.44 

.0002 

47.096 

.0007 

25.854 

.0016 

.822 

.0027 

.600 

47 

14 

.0000 

245.55 

.0002 

46.460 

.0008 

.661 

.0016 

.730 

.0027 

.547 

46 

15 

1.0000 

229.18 

1.0002 

45.840 

1.0008 

25.471 

1.0016 

17.639 

1.0028 

13.494 

45 

16 

.0000 

214.86 

.0002 

45.237 

.0008 

.284 

.0016 

.549 

.0028 

.441 

44 

17 

.0000 

202.22 

.0003 

44.650 

.0008 

25.100 

.0016 

.460 

.0028 

.389 

43 

18 

.0000 

190.99 

.0003 

44.077 

.0008 

24.918 

.0017 

.372 

.0028 

.337 

42 

19 

.0000 

180.93 

.0003 

43.520 

.0008 

.739 

.0017 

.285 

.0028 

.286 

41 

20 

1.0000 

171.89 

1.0003 

. 42.976 

1.0008 

24.562 

1.0017 

17.198 

1.0029 

13.235 

40 

21 

.0000 

163.70 

.0003 

42.445 

.0008 

.388 

.0017 

.113 

.0029 

.184 

39 

22 

.0000 

156.26 

.0003 

41.928 

.0009 

.216 

.0017 

17.028 

.0029 

.134 

38 

23 

.0000 

149.47 

.0003 

41.423 

.0009 

24.047 

.0017 

16.945 

.0029 

.084 

37 

24 

.0000 

143.24 

.0003 

40.930 

.0009 

23.880 

.0018 

.862 

.0030 

13.035 

36 

25 

1.0000 

137.51 

1.0003 

40.448 

1.0009 

23.716 

1.0018 

16.779 

1.0030 

12.985 

35 

26 

.0000 

132.22 

.0003 

39.978 

.0009 

.553 

.0018 

.698 

.0030 

.937 

34 

27 

.0000 

127.33 

.0003 

39.519 

.0009 

.393 

.0018 

.618 

.0030 

.888 

33 

28 

.0000 

122.78 

.0003 

39.070 

.0009 

.235 

.0018 

.538 

.0030 

.840 

32 

29 

.0000 

118.54 

.0003 

38.631 

.0009 

23.079 

.0019 

.459 

.0031 

.793 

31 

30 

1.0000 

114.59 

1.0003 

38.202 

1.0010 

22.926 

1.0019 

16.380 

1.0031 

12.745 

30 

31 

.0000 

110.90 

.0004 

37.782 

.0010 

.774 

.0019 

.303 

.0031 

.699 

29 

32 

.0000 

107.43 

.0004 

37.371 

.0010 

.624 

.0019 

.226 

.0031 

.652 

28 

33 

.0000 

104.18 

.0004 

36.970 

.0010 

.476 

.0019 

.150 

.0032 

.606 

27 

34 

.0000 

101.11 

.0004 

36.576 

.0010 

.330 

.0019 

.075 

.0032 

.560 

26 

35 

1.0001 

98.223 

1.0004 

36.191 

1.0010 

22.187 

1.0020 

16.000 

1.0032 

12.514 

25 

36 

.0001 

95.495 

.0004 

35.815 

.0010 

22.044 

.0020 

15.926 

.0032 

.469 

24 

37 

.0001 

92.914 

.0004 

35.445 

.0010 

21.904 

.0020 

.853 

.0033 

.424 

23 

38 

.0001 

90.469 

.0004 

35.084 

.0011 

.766 

.0020 

.780 

.0033 

.379 

22 

39 

.0001 

88.149 

.0004 

34.730 

.0011 

.629 

.0020 

.708 

.0033 

.335 

21 

40 

1.0001 

85.946 

1.0004 

34.382 

1.0011 

21.494 

1.0021 

15.637 

1.0033 

12.291 

20 

41 

.0001 

83.849 

.0004 

34.042 

.0011 

.360 

.0021 

.566 

.0034 

.248 

19 

42 

.0001 

81.853 

.0004 

33.708 

.0011 

.229 

.0021 

.496 

.0034 

.204 

18 

43 

.0001 

79.950 

.0004 

33.381 

.0011 

21.098 

.0021 

.427 

.0034 

.161 

17 

44 

.0001 

78.133 

.0005 

33.060 

.0011 

20.970 

.0021 

.358 

.0034 

.119 

16 

45 

1.0001 

76.397 

1.0005 

32.746 

1.0012 

20.843 

1.0021 

15.290 

1.0034 

12.076 

15 

46 

.0001 

74.736 

.0005 

32.437 

.0012 

.717 

.0022 

.222 

.0035 

12.034 

14 

47 

.0001 

73.146 

.0005 

32.134 

.0012 

.593 

.0022 

.155 

.0035 

11.992 

13 

48 

.0001 

71.622 

.0005 

31.836 

.0012 

.471 

.0022 

.089 

.0035 

.951 

12 

49 

.0001 

70.160 

.0005 

31.544 

.0012 

.350 

.0022 

15.023 

.0035 

.909 

11 

50 

1.0001 

68.757 

1.0005 

31.258 

1.0012 

20.230 

1.0022 

14.958 

1.0036 

11.868 

10 

51 

.0001 

67.409 

.0005 

30.976 

.0012 

20.112 

.0023 

.893 

.0036 

.828 

9 

52 

.0001 

66.113 

.0005 

30.700 

.0013 

19.995 

.0023 

.829 

.0036 

.787 

8 

53 

.0001 

64.866 

.0005 

30.428 

.0013 

.880 

.0023 

.766 

.0036 

.747 

7 

54 

.0001 

63.665 

.0006 

30.161 

.0013 

.766 

.0023 

.703 

.0037 

.707 

6 

55 

1.0001 

62.507 

1.0006 

29.899 

1.0013 

19.653 

1.0023 

14.640 

1.0037 

11.668 

5 

56 

.0001 

61.391 

.0006 

29.641 

.0013 

.541 

.0024 

.578 

.0037 

.628 

4 

57 

.0001 

60.314 

.0006 

29.388 

.0013 

.431 

.0024 

.517 

.0037 

.589 

3 

58 

.0001 

59.274 

.0006 

29.139 

.0013 

.322 

.0024 

.456 

.0038 

.551 

2 

59 

.0001 

58.270 

.0006 

28.894 

.0014 

.214 

.0024 

.395 

.0038 

.512 

1 

60 

1.0002 

57.299 

1.0006 

28.654 

1.0014 

19.107 

1.0024 

14.336 

1.0038 

11.474 

0 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 



89° 

o 

oo 

00 

o 

l> 

CO 

86° 

85° 



[ 106 ] 




























































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



5 

° 

6 

o 

7° 

8° 

£ 



/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.0038 

11.474 

1.0055 

9.5668 

1.0075 

8.2055 

1.0098 

7.1853 

1.0125 

6.3925 

60 

1 

.0038 

.436 

.0055 

.5404 

.0075 

.1861 

.0099 

.1705 

.0125 

.3807 

59 

2 

.0039 

.398 

.0056 

.5141 

.0076 

.1668 

.0099 

.1557 

.0126 

.3691 

58 

3 

.0039 

.360 

.0056 

.4880 

.0076 

.1476 

.0100 

.1410 

.0126 

.3574 

57 

4 

.0039 

.323 

.0056 

.4620 

.0077 

.1285 

.0100 

.1263 

.0127 

.3458 

56 

5 

1.0039 

11.286 

1.0057 

9.4362 

1.0077 

8.1095 

1.0100 

7.1117 

1.0127 

6.3343 

55 

6 

.0040 

.249 

.0057 

.4105 

.0077 

.0905 

.0101 

.0972 

.0127 

.3228 

54 

7 

.0040 

.213 

.0057 

.3850 

.0078 

.0717 

.0101 

.0827 

.0128 

.3113 

53 

8 

.0040 

.176 

.0058 

.3596 

.0078 

.0529 

.0102 

.0683 

.0128 

.2999 

52 

9 

.0041 

.140 

.0058 

.3343 

.0078 

.0342 

.0102 

.0539 

.0129 

.2885 

51 

10 

1.0041 

11.105 

1.0058 

9.3092 

1.0079 

8.0156 

1.0102 

7.0396 

1.0129 

6.2772 

50 

11 

.0041 

.069 

.0059 

.2842 

.0079 

7.9971 

.0103 

.0254 

.0130 

.2659 

49 

12 

.0041 

11.034 

.0059 

.2593 

.0079 

.9787 

.0103 

7.0112 

.0130 

.2546 

48 

13 

.0042 

10.998 

.0059 

.2346 

.0080 

.9604 

.0104 

6.9971 

.0131 

.2434 

47 

14 

.0042 

.963 

.0059 

.2100 

.0080 

.9422 

.0104 

.9830 

.0131 

.2323 

46 

15 

1.0042 

10.929 

1.0060 

9.1855 

1.0081 

7.9240 

1.0105 

6.9690 

1.0132 

6.2211 

45 

16 

.0042 

.894 

.0060 

.1612 

.0081 

.9059 

.0105 

.9550 

.0132 

.2100 

44 

17 

.0043 

.860 

.0060 

.1370 

.0081 

.8879 

.0105 

.9411 

.0133 

.1990 

43 

18 

.0043 

.826 

.0061 

.1129 

.0082 

.8700 

.0106 

.9273 

.0133 

.1880 

42 

19 

.0043 

.792 

.0061 

.0890 

.0082 

.8522 

.0106 

.9135 

.0134 

.1770 

41 

20 

1.0043 

10.758 

1.0061 

9.0652 

1.0082 

7.8344 

1.0107 

6.8998 

1.0134 

6.1661 

40 

21 

.0044 

.725 

.0062 

.0415 

.0083 

.8168 

.0107 

.8861 

.0135 

.1552 

39 

22 

.0044 

.692 

.0062 

9.0179 

.0083 

.7992 

.0108 

.8725 

.0135 

.1443 

38 

23 

.0044 

.659 

.0062 

8.9944 

.0084 

.7817 

.0108 

.8589 

.0136 

.1335 

37 

24 

.0045 

.626 

.0063 

.9711 

.0084 

.7642 

.0108 

.8454 

.0136 

.1227 

36 

25 

1.0045 

10.593 

1.0063 

8.9479 

1.0084 

7.7469 

1.0109 

6.8320 

1.0137 

6.1120 

35 

26 

.0045 

.561 

.0063 

.9248 

.0085 

.7296 

.0109 

.8186 

.0137 

.1013 

34 

27 

.0045 

.529 

.0064 

.9019 

.0085 

.7124 

.0110 

.8052 

.0138 

.0906 

33 

28 

.0046 

.497 

.0064 

.8790 

.0086 

.6953 

.0110 

.7919 

.0138 

.0800 

32 

29 

.0046 

.465 

.0064 

.8563 

.0086 

.6783 

.0111 

.7787 

.0139 

.0694 

31 

30 

1.0046 

10.433 

1.0065 

8.8337 

1.0086 

7.6613 

1.0111 

6.7655 

1.0139 

6.0589 

30 

31 

.0047 

.402 

.0065 

.8112 

.0087 

.6444 

.0112 

.7523 

.0140 

.0483 

29 

32 

.0047 

.371 

.0065 

.7888 

.0087 

.6276 

.0112 

.7392 

.0140 

.0379 

28 

33 

.0047 

.340 

.0066 

.7665 

.0087 

.6109 

.0112 

.7262 

.0141 

.0274 

27 

34 

.0047 

.309 

.0066 

.7444 

.0088 

.5942 

.0113 

.7132 

.0141 

.0170 

26 

35 

1.0048 

10.278 

1.0066 

8.7223 

1.0088 

7.5776 

1.0113 

6.7003 

1.0142 

6.0067 

25 

36 

.0048 

.248 

.0067 

.7004 

.0089 

.5611 

.0114 

.6874 

.0142 

5.9963 

24 

37 

.0048 

.217 

.0067 

.6786 

.0089 

.5446 

.0114 

.6745 

.0143 

.9860 

23 

38 

.0049 

.187 

.0067 

.6569 

.0089 

.5282 

.0115 

.6618 

.0143 

.9758 

22 

39 

.0049 

.157 

.0068 

.6353 

.0090 

.5119 

.0115 

.6490 

.0144 

.9656 

21 

40 

1.0049 

10.128 

1.0068 

8.6138 

1.0090 

7.4957 

1.0116 

6.6363 

1.0144 

5.9554 

20 

41 

.0049 

.098 

.0068 

.5924 

.0091 

.4795 

.0116 

.6237 

.0145 

.9452 

19 

42 

.0050 

.068 

.0069 

.5711 

.0091 

.4635 

.0116 

.6111 

.0145 

.9351 

18 

43 

.0050 

.039 

.0069 

.5500 

.0091 

.4474 

.0117 

.5986 

.0146 

.9250 

17 

44 

.0050 

10.010 

.0069 

.5289 

.0092 

.4315 

.0117 

.5861 

.0146 

.9150 

16 

45 

1.0051 

9.9812 

1.0070 

8.5079 

1.0092 

7.4156 

1.0118 

6.5736 

1.0147 

5.9049 

15 

46 

.0051 

.9525 

.0070 

.4871 

.0093 

.3998 

.0118 

.5612 

.0147 

.8950 

14 

47 

.0051 

.9239 

.0070 

.4663 

.0093 

.3840 

.0119 

.5489 

.0148 

.8850 

13 

48 

.0051 

.8955 

.0071 

.4457 

.0093 

.3684 

.0119 

.5366 

.0148 

.8751 

12 

49 

.0052 

.8672 

.0071 

.4251 

.0094 

.3527 

.0120 

.5243 

.0149 

.8652 

11 

50 

1.0052 

9.8391 

1.0072 

8.4047 

1.0094 

7.3372 

1.0120 

6.5121 

1.0149 

5.8554 

10 

51 

.0052 

.8112 

.0072 

.3843 

.0095 

.3217 

.0120 

.4999 

.0150 

.8456 

9 

52 

.0053 

.7834 

.0072 

.3641 

.0095 

.3063 

.0121 

.4878 

.0150 

.8358 

8 

53 

.0053 

.7558 

.0073 

.3439 

.0095 

.2909 

.0121 

.4757 

.0151 

.8261 

7 

54 

.0053 

.7283 

.0073 

.3238 

.0096 

.2757 

.0122 

.4637 

.0151 

.8164 

6 

55 

1.0054 

9.7010 

1.0073 

8.3039 

1.0096 

7.2604 

1.0122 

6.4517 

1.0152 

5.8067 

5 

56 

.0054 

.6739 

.0074 

.2840 

.0097 

.2453 

.0123 

.4398 

.0152 

.7970 

4 

57 

.0054 

.6469 

.0074 

.2642 

.0097 

.2302 

.0123 

.4279 

.0153 

.7874 

3 

58 

.0054 

.6200 

.0074 

.2446 

.0097 

.2152 

.0124 

.4160 

.0153 

.7778 

2 

59 

.0055 

.5933 

.0075 

.2250 

.0098 

.2002 

.0124 

.4042 

.0154 

.7683 

1 

60 

1.0055 

9.5668 

1.0075 

8.2055 

1.0098 

7.1853 

1.0125 

6.3925 

1.0154 

5.7588 

0 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 


o 

00 

83° 

82° 

0 

CO 

o 

O 

00 



[ 107 ] 
































































































COt'.OOO© »-H<NCOTtilO CDt>00©© rHMCOT^ ITS CD l> 00 © © r-4 (M CO ^ iti CDt^OO©© HC^CO^lO CD GO © © H <M CO tJH lO CD 00 © © 

▼H HHHHH hhhhn NMINWN <N<MC^C^C0 CO CO CO CO CO CO CO CO CO ^ rfl ^ Tt^ ^ ^ © IQ iO iO *0 lO lO *D *C lO © 


XI. FIVE-PLACE VALUES: SECANT AND COSECANT 


h* 

© 

O 

o 

▼H 

tH 

12° 

13° 

14° 


Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


1.0154 

5.7588 

1.0187 

5.2408 

1.0223 

4.8097 

1.0263 

4.4454 

1.0306 

4.1336 

60 

.0155 

.7493 

.0188 

.2330 

.0224 

.8032 

.0264 

.4398 

.0307 

.1287 

59 

.0155 

.7398 

.0188 

.2252 

.0225 

.7966 

.0264 

.4342 

.0308 

.1239 

58 

.0156 

.7304 

.0189 

.2174 

.0225 

.7901 

.0265 

.4287 

.0308 

.1191 

57 

.0156 

.7210 

.0189 

.2097 

.0226 

.7836 

.0266 

.4231 

.0309 

- .1144 

56 

1.0157 

5.7117 

1.0190 

5.2019 

1.0227 

4.7771 

1.0266 

4.4176 

1.0310 

4.1096 

55 

.0157 

.7023 

.0191 

.1942 

.0227 

.7706 

.0267 

.4121 

.0311 

.1048 

54 

.0158 

.6930 

.0191 

.1865 

.0228 

.7641 

.0268 

.4066 

.0311 

.1001 

53 

.0158 

.6838 

.0192 

.1789 

.0228 

.7577 

.0269 

.4011 

.0312 

.0954 

52 

.0159 

.6745 

.0192 

.1712 

.0229 

.7512 

.0269 

.3956 

.0313 

.0906 

51 

1.0160 

5.6653 

1.0193 

5.1636 

1.0230 

4.7448 

1.0270 

4.3901 

1.0314 

4.0859 

50 

.0160 

.6562 

.0194 

.1560 

.0230 

.7384 

.0271 

.3847 

.0314 

.0812 

49 

.0161 

.6470 

.0194 

.1484 

.0231 

.7321 

.0271 

.3792 

.0315 

.0765 

48 

.0161 

.6379 

.0195 

.1409 

.0232 

.7257 

.0272 

.3738 

.0316 

.0718 

47 

.0162 

.6288 

.0195 

.1333 

.0232 

.7194 

.0273 

.3684 

.0317 

.0672 

46 

1.0162 

5.6198 

1.0196 

5.1258 

1.0233 

4.7130 

1.0273 

4.3630 

1.0317 

4.0625 

45 

.0163 

.6107 

.0197 

.1183 

.0234 

.7067 

.0274 

.3576 

.0318 

.0579 

44 

.0163 

.6017 

.0197 

.1109 

.0234 

.7004 

.0275 

.3522 

.0319 

.0532 

43 

.0164 

.5928 

.0198 

.1034 

.0235 

.6942 

.0276 

.3469 

.0320 

.0486 

42 

.0164 

.5838 

.0198 

.0960 

.0236 

.6879 

.0276 

.3415 

.0321 

.0440 

41 

1.0165 

5.5749 

1.0199 

5.0886 

1.0236 

4.6817 

1.0277 

4.3362 

1.0321 

4.0394 

40 

.0165 

.5660 

.0199 

.0813 

.0237 

.6755 

.0278 

.3309 

.0322 

.0348 

39 

.0166 

.5572 

.0200 

.0739 

.0238 

.6693 

.0278 

.3256 

.0323 

.0302 

38 

.0166 

.5484 

.0201 

.0666 

.0238 

.6631 

.0279 

.3203 

.0324 

.0256 

37 

.0167 

.5396 

.0201 

.0593 

.0239 

.6569 

.0280 

.3150 

.0324 

.0211 

36 

1.0168 

5.5308 

1.0202 

5.0520 

1.0240 

4.6507 

1.0281 

4.3098 

1.0325 

4.0165 

35 

.0168 

.5221 

.0202 

.0447 

.0240 

.6446 

.0281 

.3045 

.0326 

.0120 

34 

.0169 

.5134 

.0203 

.0375 

.0241 

.6385 

.0282 

.2993 

.0327 

.0075 

33 

.0169 

.5047 

.0204 

.0302 

.0241 

.6324 

.0283 

.2941 

.0327 

4.0029 

32 

.0170 

.4960 

.0204 

.0230 

.0242 

.6263 

.0283 

.2889 

.0328 

3.9984 

31 

1.0170 

5.4874 

1.0205 

5.0159 

1.0243 

4.6202 

1.0284 

4.2837 

1.0329 

3.9939 

30 

.0171 

.4788 

.0205 

.0087 

.0243 

.6142 

.0285 

.2785 

.0330 

.9894 

29 

.0171 

.4702 

.0206 

5.0016 

.0244 

.6081 

.0286 

.2733 

.0331 

.9850 

28 

.0172 

.4617 

.0207 

4.9944 

.0245 

.6021 

.0286 

.2681 

.0331 

.9805 

27 

.0173 

.4532 

.0207 

.9873 

.0245 

.5961 

.0287 

.2630 

.0332 

.9760 

26 

1.0173 

5.4447 

1.0208 

4.9803 

1.0246 

4.5901 

1.0288 

4.2579 

1.0333 

3.9716 

25 

.0174 

.4362 

.0209 

.9732 

.0247 

.5841 

.0288 

.2527 

.0334 

.9672 

24 

.0174 

.4278 

.0209 

.9662 

.0247 

.5782 

.0289 

.2476 

.0334 

.9627 

23 

.0175 

.4194 

.0210 

.9591 

.0248 

.5722 

.0290 

.2425 

.0335 

.9583 

22 

.0175 

.4110 

.0210 

.9521 

.0249 

.5663 

.0291 

.2375 

.0336 

.9539 

21 

1.0176 

5.4026 

1.0211 

4.9452 

1.0249 

4.5604 

1.0291 

4.2324 

1.0337 

3.9495 

20 

.0176 

.3943 

.0212 

.9382 

.0250 

.5545 

.0292 

.2273 

.0338 

.9451 

19 

.0177 

.3860 

.0212 

.9313 

.0251 

.5486 

.0293 

.2223 

.0338 

.9408 

18 

.0178 

.3777 

.0213 

.9244 

.0251 

.5428 

.0294 

.2173 

.0339 

.9364 

17 

.0178 

.3695 

.0213 

.9175 

.0252 

.5369 

.0294 

.2122 

.0340 

.9320 

16 

1.0179 

5.3612 

1.0214 

4.9106 

1.0253 

4.5311 

1.0295 

4.2072 

1.0341 

3.9277 

15 

.0179 

.3530 

.0215 

.9037 

.0253 

.5253 

.0296 

.2022 

.0342 

.9234 

14 

.0180 

.3449 

.0215 

.8969 

.0254 

.5195 

.0297 

.1973 

.0342 

.9190 

13 

.0180 

.3367 

.0216 

.8901 

.0255 

.5137 

.0297 

.1923 

.0343 

.9147 

12 

.0181 

.3286 

.0217 

.8833 

.0256 

.5079 

.0298 

.1873 

.0344 

.9104 

11 

1.0181 

5.3205 

1.0217 

4.8765 

1.0256 

4.5022 

1.0299 

4.1824 

1.0345 

3.9061 

10 

.0182 

.3124 

.0218 

.8697 

.0257 

.4964 

.0299 

.1774 

.0346 

.9018 

9 

.0183 

.3044 

.0218 

.8630 

.0258 

.4907 

.0300 

.1725 

.0346 

.8976 

8 

.0183 

.2963 

.0219 

.8563 

.0258 

.4850 

.0301 

.1676 

.0347 

.8933 

7 

.0184 

.2883 

.0220 

.8496 

.0259 

.4793 

.0302 

.1627 

.0348 

.8890 

6 

1.0184 

5.2804 

1.0220 

4.8429 

1.0260 

4.4736 

1.0302 

4.1578 

1.0349 

3.8848 

5 

.0185 

.2724 

.0221 

.8362 

.0260 

.4679 

.0303 

.1529 

.0350 

.8806 

4 

.0185 

.2645 

.0222 

.8296 

.0261 

.4623 

.0304 

.1481 

.0350 

.8763 

3 

.0186 

.2566 

.0222 

.8229 

.0262 

.4566 

.0305 

.1432 

.0351 

.8721 

2 

.0187 

.2487 

.0223 

.8163 

.0262 

.4510 

.0305 

.1384 

.0352 

.8679 

1 

1.0187 

5.2408 

1.0223 

4.8097 

1.0263 

4.4454 

1.0306 

4.1336 

1.0353 

3.8637 

0 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 

79° 

o 

00 

77° 

76° 

75° 



[ 108 ] 




























































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



15° 

16° 

17° 

o 

co 

19° 



Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.0353 

3.8637 

1.0403 

3.6280 

1.0457 

3.4203 

1.0515 

3.2361 

1.0576 

3.0716 

60 

1 

.0354 

.8595 

.0404 

.6243 

.0458 

.4171 

.0516 

.2332 

.0577 

.0690 

59 

; 2 

.0354 

.8553 

.0405 

.6206 

.0459 

.4138 

.0517 

.2303 

.0578 

0664 

58 

3 

.0355 

.8512 

.0406 

.6169 

.0460 

.4106 

.0518 

.2274 

.0579 

.0638 

57 

4 

.0356 

.8470 

.0406 

.6133 

.0461 

.4073 

.0519 

.2245 

.0580 

.0612 

56 

5 

1.0357 

3.8428 

1.0407 

3.6097 

1.0462 

3.4041 

1.0520 

3.2217 

1.0582 

3.0586 

55 

6 

.0358 

.8387 

.0408 

.6060 

.0463 

.4009 

.0521 

.2188 

.0583 

.0561 

54 

, 7 

.0358 

.8346 

.0409 

.6024 

.0463 

.3977 

.0522 

.2159 

.0584 

.0535 

53 

8 

.0359 

.8304 

.0410 

.5988 

.0464 

.3945 

.0523 

.2131 

.0585 

.0509 

52 

9 

.0360 

.8263 

.0411 

.5951 

.0465 

.3913 

.0524 

.2102 

.0586 

.0484 

51 

10 

1.0361 

3.8222 

1.0412 

3.5915 

1.0466 

3.3881 

1.0525 

3.2074 

1.0587 

3.0458 

50 

11 

.0362 

.8181 

.0413 

.5879 

.0467 

.3849 

.0526 

.2045 

.0588 

.0433 

49 

12 

.0363 

.8140 

.0413 

.5843 

.0468 

.3817 

.0527 

.2017 

.0589 

.0407 

48 

13 

.0363 

.8100 

.0414 

.5808 

.0469 

.3785 

.0528 

.1989 

.0590 

.0382 

47 

14 

.0364 

.8059 

.0415 

.5772 

.0470 

.3754 

.0529 

.1960 

.0591 

.0357 

46 

15 

1.0365 

3.8018 

1.0416 

3.5736 

1.0471 

3.3722 

1.0530 

3.1932 

1.0592 

3.0331 

45 

16 

.0366 

.7978 

.0417 

.5700 

.0472 

.3691 

.0531 

.1904 

.0593 

.0306 

44 

17 

.0367 

.7937 

.0418 

.5665 

.0473 

.3659 

.0532 

.1876 

.0594 

.0281 

43 

18 

.0367 

.7897 

.0419 

.5629 

.0474 

.3628 

.0533 

.1848 

.0595 

.0256 

42 

19 

.0368 

.7857 

.0420 

.5594 

.0475 

.3596 

.0534 

.1820 

.0597 

.0231 

41 

20 

1.0369 

3.7817 

1.0421 

3.5559 

1.0476 

3.3565 

1.0535 

3.1792 

1.0598 

3.0206 

40 

21 

.0370 

.7777 

.0421 

.5523 

.0477 

.3534 

.0536 

.1764 

.0599 

.0181 

39 

22 

.0371 

.7737 

.0422 

.5488 

.0478 

.3502 

.0537 

.1736 

.0600 

.0156 

38 

23 

.0372 

.7697 

.0423 

.5453 

.0479 

.3471 

.0538 

.1708 

.0601 

.0131 

37 

24 

.0372 

.7657 

.0424 

.5418 

.0480 

.3440 

.0539 

.1681 

.0602 

.0106 

36 

25 

1.0373 

3.7617 

1.0425 

3.5383 

1.0480 

3.3409 

1.0540 

3.1653 

1.0603 

3.0081 

35 

26 

.0374 

.7577 

.0426 

.5348 

.0481 

.3378 

.0541 

.1625 

.0604 

.0056 

34 

27 

.0375 

.7538 

.0427 

.5313 

.0482 

.3347 

.0542 

.1598 

.0605 

.0031 

33 

28 

.0376 

.7498 

.0428 

.5279 

.0483 

.3317 

.0543 

.1570 

.0606 

3.0007 

32 

29 

.0377 

.7459 

.0429 

.5244 

.0484 

.3286 

.0544 

.1543 

.0607 

2.9982 

31 

30 

1.0377 

3.7420 

1.0429 

3.5209 

1.0485 

3.3255 

1.0545 

3.1515 

1.0608 

2.9957 

30 

31 

.0378 

.7381 

.0430 

.5175 

.0486 

.3224 

.0546 

.1488 

.0610 

.9933 

29 

32 

.0379 

.7341 

.0431 

.5140 

.0487 

.3194 

.0547 

.1461 

.0611 

.9908 

28 

33 

.0380 

.7302 

.0432 

.5106 

.0488 

.3163 

.0548 

.1433 

.0612 

.9884 

27 

34 

.0381 

.7263 

.0433 

.5072 

.0489 

.3133 

.0549 

.1406 

.0613 

.9859 

26 

35 

1.0382 

3.7225 

1.0434 

3.5037 

1.0490 

3.3102 

1.0550 

3.1379 

1.0614 

2.9835 

25 

36 

.0382 

.7186 

.0435 

.5003 

.0491 

.3072 

.0551 

.1352 

.0615 

.9811 

24 

37 

.0383 

.7147 

.0436 

.4969 

.0492 

.3042 

.0552 

.1325 

.0616 

.9786 

23 

38 

.0384 

.7108 

.0437 

.4935 

.0493 

.3012 

.0553 

.1298 

.0617 

.9762 

22 

39 

.0385 

.7070 

.0438 

.4901 

.0494 

.2981 

.0554 

.1271 

.0618 

.9738 

21 

40 

1.0386 

3.7032 

1.0439 

3.4867 

1.0495 

3.2951 

1.0555 

3.1244 

1.0619 

2.9713 

20 

41 

.0387 

.6993 

.0439 

.4833 

.0496 

.2921 

.0556 

.1217 

.0621 

.9689 

19 

42 

.0388 

.6955 

.0440 

.4799 

.0497 

.2891 

.0557 

.1190 

.0622 

.9665 

18 

43 

.0388 

.6917 

.0441 

.4766 

.0498 

.2861 

.0558 

.1163 

.0623 

.9641 

17 

44 

.0389 

.6879 

.0442 

.4732 

.0499 

.2831 

.0559 

.1137 

.0624 

.9617 

16 

45 

1.0390 

3.6840 

1.0443 

3.4699 

1.0500 

3.2801 

1.0560 

3.1110 

1.0625 

2.9593 

15 

46 

.0391 

.6803 

.0444 

.4665 

.0501 

.2772 

.0561 

.1083 

.0626 

.9569 

14 

47 

.0392 

.6765 

.0445 

.4632 

.0502 

.2742 

.0563 

.1057 

.0627 

.9545 

13 

48 

.0393 

.6727 

.0446 

.4598 

.0503 

.2712 

.0564 

.1030 

.0628 

.9521 

12 

49 

.0394 

.6689 

.0447 

.4565 

.0504 

.2683 

.0565 

.1004 

.0629 

.9498 

11 

50 

1.0394 

3.6652 

1.0448 

3.4532 

1.0505 

3.2653 

1.0566 

3.0977 

1.0631 

2.9474 

10 

51 

.0395 

.6614 

.0449 

.4499 

.0506 

.2624 

.0567 

.0951 

.0632 

.9450 

9 

52 

.0396 

.6576 

.0450 

.4465 

.0507 

.2594 

.0568 

.0925 

.0633 

.9426 

8 

53 

.0397 

.6539 

.0450 

.4432 

.0508 

.2565 

.0569 

.0898 

.0634 

.9403 

7 

54 

.0398 

.6502 

.0451 

.4399 

.0509 

.2535 

.0570 

.0872 

.0635 

.9379 

6 

55 

1.0399 

3.6465 

1.0452 

3.4367 

1.0510 

3.2506 

1.0571 

3.0846 

1.0636 

2.9355 

5 

56 

.0400 

.6427 

.0453 

.4334 

.0511 

.2477 

.0572 

.0820 

.0637 

.9332 

4 

57 

.0400 

.6390 

.0454 

.4301 

.0512 

.2448 

.0573 

.0794 

.0638 

.9308 

3 

58 

.0401 

.6353 

.0455 

.4268 

.0513 

.2419 

.0574 

.0768 

.0640 

.9285 

2 

59 

.0402 

.6316 

.0456 

.4236 

.0514 

.2390 

.0575 

.0742 

.0641 

.9261 

1 

60 

1.0403 

3.6280 

1.0457 

3.4203 

1.0515 

3.2361 

1.0576 

3.0716 

1.0642 

2.9238 

0 


Csc 

Sec 

Csc 

Sec 

Csc 1 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 


74° 

73 

o 

72 

o 

71 

0 

70 

0 



[ 109 ] 






















































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



20° 

21° 

22° 

23° 

24° 

I | 

/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1 

2 

3 

4 

5 

6 

' 7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

1.0642 

.0643 

.0644 

.0645 

.0646 

1.0647 

.0649 

.0650 

.0651 

.0652 

1.0653 

2.9238 

.9215 

.9191 

..9168 

.9145 

2.9122 

.9099 

.9075 

.9052 

.9029 

2.9006 

1.0711 

.0713 

.0714 

.0715 

.0716 

1.0717 

.0719 

.0720 

.0721 

.0722 

1.0723 

2.7904 

.7883 

.7862 

.7841 

.7820 

2.7799 

.7778 

.7757 

.7736 

.7715 

2.7695 

1.0785 

.0787 

.0788 

.0789 

.0790 

1.0792 

.0793 

.0794 

.0796 

.0797 

1.0798 

2.6695 

.6675 

.6656 

.6637 

.6618 

2.6599 

.6580 

.6561 

.6542 

.6523 

2.6504 

1.0864 

.0865 

.0866 

.0868 

.0869 

1.0870 

.0872 

.0873 

.0874 

.0876 

1.0877 

2.5593 

.5576 

.5558 

.5541 

.5523 

2.5506 

.5488 

.5471 

.5454 

.5436 

2.5419 

1.0946 

.0948 

.0949 

.0951 

.0952 

1.0953 

.0955 

.0956 

.0958 

.0959 

1.0961 

2.4586 

.4570 

.4554 

.4538 

.4522 

2.4506 

.4490 

.4474 

.4458 

.4442 

2.4426 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

.0654 

.0655 

.0657 

.0658 

1.0659 

.0660 

.0661 

.0662 

.0663 

1.0665 

.8983 

.8960 

.8938 

.8915 

2.8892 

.8869 

.8846 

.8824 

.8801 

2.8779 

.0725 

.0726 

.0727 

.0728 

1.0730 

.0731 

.0732 

.0733 

.0734 

1.0736 

.7674 

.7653 

.7632 

.7612 

2.7591 

.7570 

.7550 

.7529 

.7509 

2.7488 

.0799 

.0801 

.0802 

.0803 

1.0804 

.0806 

.0807 

.0808 

.0810 

1.0811 

.6485 

.6466 

.6447 

.6429 

2.6410 

.6391 

.6372 

.6354 

.6335 

2.6316 

.0878 
, .0880 
.0881 
.0883 
1.0884 

.0885 

.0887 

.0888 

.0889 

1.0891 

.5402 

.5384 

.5367 

.5350 

2.5333 

.5316 

.5299 

.5282 

.5264 

2.5247 

.0962 

.0963 

.0965 

.0966 

1.0968 

.0969 

.0971 

.0972 

.0974 

1.0975 

.4411 

.4395 

.4379 

.4363 

2.4348 

.4332 

.4316 

.4300 

.4285 

2.4269 

49 

48 

47 

46 

45 

44 

43 

42 

41 

4° 

39 

38 

37 

1 36 

35 

34 

33 

32 

31 

30 

29 

28 

27 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

n 

10 

9 

3 

7 

6 

5 

4 

3 

2 

1 

0 

.0666 

.0667 

.0668 

.0669 

1.0670 

.0671 

.0673 

.0674 

.0675 

1.0676 

.8756 

.8733 

.8711 

.8688 

2.8666 

.8644 

'.8621 

.8599 

.8577 

2.8555 

.0737 

.0738 

.0739 

.0740 

1.0742 

.0743 

.0744 

.0745 

.0747 

1.0748 

.7468 

.7447 

.7427 

.7407 

2.7386 

.7366 

.7346 

.7325 

.7305 

2.7285 

.0812 

.0814 

.0815 

.0816 

1.0817 

.0819 

.0820 

.0821 

.0823 

1.0824 

.6298 

.6279 

.6260 

.6242 

2.6223 

.6205 

.6186 

.6168 

.6150 

2.6131 

.0892 

.0893 

.0895 

.0896 

1.0898 

.0899 

.0900 

.0902 

.0903 

1.0904 

.5230 

.5213 

.5196 

.5180 

2.5163 

.5146 

.5129 

.5112 

.5095 

2.5078 

.0976 

.0978 

.0979 

.0981 

1.0982 

.0984 

.0985 

.0987 

.0988 

1.0989 

.4254 

.4238 

.4222 

.4207 

2.4191 

.4176 

.4160 

.4145 

.4130 

2.4114 

.0677 

.0678 

.0680 

.0681 

1.0682 

.0683 

.0684 

.0685 

.0687 

1.0688 

.8532 

.8510 

.8488 

.8466 

2.8444 

.8422 

.8400 

.8378 

.8356 

2.8334 

.0749 

.0750 

.0752 

.0753 

1.0754 

.0755 

.0757 

.0758 

.0759 

1.0760 

.7265 

.7245 

.7225 

.7205 

2.7185 

.7165 

.7145 

.7125 

.7105 

2.7085 

.0825 

.0827 

.0828 

.0829 

1.0830 

.0832 

.0833 

.0834 

.0836 

1.0837 

.6113 

.6095 

.6076 

.6058 

2.6040 

.6022 

.6003 

.5985 

.5967 

2.5949 

.0906 

.0907 

.0909 

.0910 

1.0911 

.0913 

.0914 

.0915 

.0917 

1.0918 

.5062 

.5045 

.5028 

.5012 

2.4995 

.4978 

.4962 

.4945 

.4928 

2.4912 

.0991 

.0992 

.0994 

.0995 

1.0997 

.0998 

.1000 

.1001 

.1003 

1.1004 

.4099 ! 
.4083 
.4068 
.4053 
2.4038 

.4022 

.4007 

.3992 

.3977 

2.3961 

.0689 

.0690 

.0691 

.0692 

1.0694 

.0695 

.0696 

.0697 

.0698 

1.0700 

.8312 

.8291 

.8269 

.8247 

2.8225 

.8204 

.8182 

.8161 

.8139 

2.8117 

.0761 

.0763 

.0764 

.0765 

1.0766 

.0768 

.0769 

.0770 

.0771 

1.0773 

.7065 

.7046 

.7026 

.7006 

2.6986 

.6967 

.6947 

.6927 

.6908 

2.6888 

.0838 

.0840 

.0841 

.0842 

1.0844 

.0845 

.0846 

.0848 

.0849 

1.0850 

.5931 

.5913 

.5895 

.5877 

2.5859 

.5841 

.5823 

.5805 

.5788 

2.5770 

.0920 

.0921 

.0922 

.0924 

1.0925 

.0927 

.0928 

.0929 

.0931 

1.0932 

.4895 

.4879 

.4862 

.4846 

2.4830 

.4813 

.4797 

.4780 

.4764 

2.4748 

.1006 

.1007 

.1009 

.1010 

1.1011 

.1013 

.1014 

.1016 

.1017 

1.1019 

.3946 

.3931 

.3916 

.3901 

2.3886 

.3871 

.3856 

.3841 

.3826 

2.3811 

.0701 

.0702 

.0703 

.0704 

1.0705 

.0707 

.0708 

.0709 

.0710 

1.0711 

.8096 

.8075 

.8053 

.8032 

2.8010 

.7989 

.7968 

.7947 

.7925 

2.7904 

.0774 

.0775 

.0777 

.0778 

1.0779 

.0780 

.0782 

.0783 

.0784 

1.0785 

.6869 

.6849 

.6830 

.6811 

2.6791 

.6772 

.6752 

.6733 

.6714 

2.6695 

.0852 

.0853 

.0854 

.0856 

1.0857 

.0858 

.0860 

.0861 

.0862 

1.0864 

.5752 

.5734 

.5716 

.5699 

2.5681 

.5663 

.5646 

.5628 

.5611 

2.5593 

.0934 

.0935 

.0936 

.0938 

1.0939 

.0941 

.0942 

.0944 

.0945 

1.0946 

.4731 

.4715 

.4699 

.4683 

2.4667 

.4650 

.4634 

.4618 

.4602 

2.4586 

.1020 

.1022 

.1023 

.1025 

1.1026 

.1028 

.1029 

.1031 

.1032 

1.1034 

.3796 

.3781 

.3766 

.3751 

2.3736 

.3721 

.3706 

.3692 

.3677 

2.3662 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 1 


69° 

68° 

67° 

66° 

65° 

-<7- 


[HO] 

































































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



25° 

26° 

27° 

28° 

29° 


/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.1034 

2.3662 

1.1126 

2.2812 

1.1223 

2.2027 

1.1326 

2.1301 

1.1434 

2.0627 

60 

1 

.1035 

.3647 

.1128 

.2798 

.1225 

.2014 

.1327 

.1289 

.1435 

.0616 

59 

2 

.1037 

.3633 

.1129 

.2785 

.1227 

.2002 

.1329 

.1277 

.1437 

.0605 

58 

3 

.1038 

.3618 

.1131 

.2771 

.1228 

.1989 

.1331 

.1266 

.1439 

.0594 

57 

4 

.1040 

.3603 

.1132 

.2757 

.1230 

.1977 

.1333 

.1254 

.1441 

.0583 

56 

5 

1.1041 

2.3588 

1.1134 

2.2744 

1.1232 

2.1964 

1.1334 

2.1242 

1.1443 

2.0573 

55 

6 

.1043 

.3574 

.1136 

.2730 

.1233 

.1952 

.1336 

.1231 

.1445 

.0562 

54 

7 

.1044 

.3559 

.1137 

.2717 

.1235 

.1939 

.1338 

.1219 

.1446 

.0551 

53 

8 

.1046 

.3545 

.1139 

.2703 

.1237 

.1927 

.1340 

.1208 

.1448 

.0540 

52 

9 

.1047 

.3530 

.1140 

.2690 

.1238 

.1914 

.1342 

.1196 

.1450 

.0530 

51 

10 

1.1049 

2.3515 

1.1142 

2.2677 

1.1240 

2.1902 

1.1343 

2.1185 

1.1452 

2.0519 

50 

11 

.1050 

.3501 

.1143 

.2663 

.1242 

.1890 

.1345 

.1173 

.1454 

.0508 

49 

12 

.1052 

.3486 

.1145 

.2650 

.1243 

.1877 

.1347 

.1162 

.1456 

.0498 

48 

13 

.1053 

.3472 

.1147 

.2636 

.1245 

.1865 

.1349 

.1150 

.1458 

.0487 

47 

14 

.1055 

.3457 

.1148 

.2623 

.1247 

.1852 

.1350 

.1139 

.1460 

.0476 

46 

15 

1.1056 

2.3443 

1.1150 

2.2610 

1.1248 

2.1840 

1.1352 

2.1127 

1.1461 

2.0466 

45 

16 

.1058 

.3428 

.1151 

.2596 

.1250 

.1828 

.1354 

.1116 

.1463 

.0455 

44 

17 

.1059 

.3414 

.1153 

.2583 

.1252 

.1815 

.1356 

.1105 

.1465 

.0445 

43 

18 

.1061 

.3400 

.1155 

.2570 

.1253 

.1803 

.1357 

.1093 

.1467 

.0434 

42 

19 

.1062 

.3385 

.1156 

.2556 

.1255 

.1791 

.1359 

.1082 

.1469 

.0423 

41 

20 

1.1064 

2.3371 

1.1158 

2.2543 

1.1257 

2.1779 

1.1361 

2.1070 

1.1471 

2.0413 

40 

21 

.1066 

.3356 

.1159 

.2530 

.1259 

.1766 

.1363 

.1059 

.1473 

.0402 

39 

22 

.1067 

.3342 

.1161 

.2517 

.1260 

.1754 

.1365 

.1048 

.1474 

.0392 

38 

23 

.1069 

.3328 

.1163 

.2504 

.1262 

.1742 

.1366 

.1036 

.1476 

.0381 

37 

24 

.1070 

.3314 

.1164 

.2490 

.1264 

.1730 

.1368 

.1025 

.1478 

.0371 

36 

25 

1.1072 

2.3299 

1.1166 

2.2477 

1.1265 

2.1718 

1.1370 

2.1014 

1.1480 

2.0360 

35 

26 

.1073 

.3285 

.1168 

.2464 

.1267 

.1705 

.1372 

.1002 

.1482 

.0350 

34 

27 

.1075 

.3271 

.1169 

.2451 

.1269 

.1693 

.1374 

.0991 

.1484 

.0339 

33 

28 

.1076 

.3257 

.1171 

.2438 

.1270 

.1681 

.1375 

.0980 

.1486 

.0329 

32 

29 

.1078 

.3242 

.1172 

.2425 

.1272 

.1669 

.1377 

.0969 

.1488 

.0318 

31 

30 

1.1079 

2.3228 

1.1174 

2.2412 

1.1274 

2.1657 

1.1379 

2.0957 

1.1490 

2.0308 

30 

31 

.1081 

.3214 

.1176 

.2399 

.1276 

.1645 

.1381 

.0946 

.1491 

.0297 

29 

32 

.1082 

.3200 

.1177 

.2385 

.1277 

.1633 

.1383 

.0935 

.1493 

.0287 

28 

33 

.1084 

.3186 

.1179 

.2372 

.1279 

.1621 

.1384 

.0924 

.1495 

.0276 

27 

34 

.1085 

.3172 

.1180 

.2359 

.1281 

.1609 

.1386 

.0913 

.1497 

.0266 

26 

35 

1.1087 

2.3158 

1.1182 

2.2346 

1.1282 

2.1596 

1.1388 

2.0901 

1.1499 

2.0256 

25 

36 

.1089 

.3144 

.1184 

.2333 

.1284 

.1584 

.1390 

.0890 

.1501 

.0245 

24 

37 

.1090 

.3130 

.1185 

.2320 

.1286 

.1572 

.1392 

.0879 

.1503 

.0235 

23 

38 

.1092 

.3115 

.1187 

.2308 

.1288 

.1560 

.1393 

.0868 

.1505 

.0225 

22 

39 

.1093 

.3101 

.1189 

.2295 

.1289 

.1549 

.1395 

.0857 

.1507 

.0214 

21 

40 

1.1095 

2.3088 

1.1190 

2.2282 

1.1291 

2.1537 

1.1397 

2.0846 

1.1509 

2.0204 

20 

41 

.1096 

.3074 

.1192 

.2269 

.1293 

.1525 

.1399 

.0835 

.1510 

.0194 

19 

42 

.1098 

.3060 

.1194 

.2256 

.1294 

.1513 

.1401 

.0824 

.1512 

.0183 

18 

43 

.1099 

.3046 

.1195 

.2243 

.1296 

.1501 

.1402 

.0813 

.1514 

.0173 

17 

44 

.1101 

.3032 

.1197 

.2230 

.1298 

.1489 

.1404 

.0802 

.1516 

.0163 

16 

45 

1.1102 

2.3018 

1.1198 

2.2217 

1.1300 

2.1477 

1.1406 

2.0791 

1.1518 

2.0152 

15 

46 

.1104 

.3004 

.1200 

.2205 

.1301 

.1465 

.1408 

.0779 

.1520 

.0142 

14 

47 

.1106 

.2990 

.1202 

.2192 

.1303 

.1453 

.1410 

.0768 

.1522 

.0132 

13 

48 

.1107 

.2976 

.1203 

.2179 

.1305 

.1441 

.1412 

.0757 

.1524 

.0122 

12 

49 

.1109 

.2962 

.1205 

.2166 

.1307 

.1430 

.1413 

.0747 

.1526 

.0112 

11 

50 

1.1110 

2.2949 

1.1207 

2.2153 

1.1308 

2.1418 

1.1415 

2.0736 

1.1528 

2.0101 

10 

51 

.1112 

.2935 

.1208 

.2141 

.1310 

.1406 

.1417 

.0725 

.1530 

.0091 

9 

52 

.1113 

.2921 

.1210 

.2128 

.1312 

.1394 

.1419 

.0714 

.1532 

.0081 

8 

53 

.1115 

.2907 

.1212 

.2115 

.1313 

.1382 

.1421 

.0703 

.1533 

.0071 

7 

54 

.1117 

.2894 

.1213 

.2103 

.1315 

.1371 

.1423 

.0692 

.1535 

.0061 

6 

55 

1.1118 

2.2880 

1.1215 

2.2090 

1.1317 

2.1359 

1.1424 

2.0681 

1.1537 

2.0051 

5 

56 

.1120 

.2866 

.1217 

.2077 

.1319 

.1347 

.1426 

.0670 

.1539 

.0040 

4 

57 

.1121 

.2853 

.1218 

.2065 

.1320 

.1336 

.1428 

.0659 

.1541 

.0030 

3 

58 

.1123 

.2839 

.1220 

.2052 

.1322 

.1324 

.1430 

.0648 

.1543 

.0020 

2 

59 

.1124 

.2825 

.1222 

.2039 

.1324 

.1312 

.1432 

.0637 

.1545 

.0010 

1 

60 

1.1126 

2.2812 

1.1223 

2.2027 

1.1326 

2.1301 

1.1434 

2.0627 

1.1547 

2.0000 

0 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 


64° 

63° 

62° 

61° 

60° 



[ 111 ] 




































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



CO 

o 

o 

31° 

32° 

33° 

34° 


/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

j 

0 

1.1547 

2.0000 

1.1666 

1.9416 

1.1792 

1.8871 

1.1924 

1.8361 

1.2062 

1.7883 

60 

1 

.1549 

1.9990 

.1668 

.9407 

.1794 

.8862 

.1926 

.8353 

.2065 

.7875 

59 

2 

.1551 

.9980 

.1670 

.9397 

.1796 

.8853 

.1928 

.8344 

.2067 

.7868 

58 1 

3 

.1553 

.9970 

.1672 

.9388 

.1798 

.8844 

.1930 

.8336 

.2069 

.7860 

57 

4 

.1555 

.9960 

.1675 

.9379 

.1800 

.8836 

.1933 

.8328 

.2072 

.7852 

56 1 

5 

1.1557 

1.9950 

1.1677 

1.9369 

1.1803 

1.8827 

1.1935 

1.8320 

1.2074 

1.7844 

55 1 

6 

.1559 

.9940 

.1679 

.9360 

.1805 

.8818 

.1937 

.8312 

.2076 

.7837 

54 

7 

.1561 

.9930 

.1681 

.9351 

.1807 

.8810 

.1939 

.8303 

.2079 

.7829 

53 

: 8 

.1563 

.9920 

.1683 

.9341 

.1809 

.8801 

.1942 

.8295 

.2081 

.7821 

52 

9 

.1565 

.9910 

.1685 

.9332 

.1811 

.8792 

.1944 

.8287 

.2084 

.7814 

51 

10 

1.1566 

1.9900 

1.1687 

1.9323 

1.1813 

1.8783 

1.1946 

1.8279 

1.2086 

1.7806 

50 1 

11 

.1568 

.9890 

.1689 

.9313 

.1815 

.8775 

.1949 

.8271 

.2088 

.7799 

49 

12 

.1570 

.9880 

.1691 

.9304 

.1818 

.8766 

.1951 

.8263 

.2091 

.7791 

48 

13 

.1572 

.9870 

.1693 

.9295 

.1820 

.8757 

.1953 

.8255 

.2093 

.7783 

47 

14 

.1574 

.9860 

.1695 

.9285 

.1822 

.8749 

.1955 

.8247 

.2096 

.7776 

46 

15 

1.1576 

1.9850 

1.1697 

1.9276 

1.1824 

1.8740 

1.1958 

1.8238 

1.2098 

1.7768 

451 

16 

.1578 

.9840 

.1699 

.9267 

.1826 

.8731 

.1960 

.8230 

.2100 

.7761 

44 

17 

.1580 

.9830 

.1701 

.9258 

.1828 

.8723 

.1962 

.8222 

.2103 

.7753 

43 

18 

.1582 

.9821 

.1703 

.9249 

.1831 

.8714 

.1964 

.8214 

.2105 

.7745 

42 

19 

.1584 

.9811 

.1705 

.9239 

.1833 

.8706 

.1967 

.8206 

.2108 

.7738 

41 

20 

1.1586 

1.9801 

1.1707 

1.9230 

1.1835 

1.8697 

1.1969 

1.8198 

1.2110 

1.7730 

401 

21 

.1588 

.9791 

.1710 

.9221 

.1837 

.8688 

.1971 

.8190 

.2112 

.7723 

39 

22 

.1590 

.9781 

.1712 

.9212 

.1839 

.8680 

.1974 

.8182 

.2115 

.7715 

38 

23 

.1592 

.9771 

.1714 

.9203 

.1842 

.8671 

.1976 

.8174 

.2117 

.7708 

37 

24 

.1594 

.9762 

.1716 

.9194 

.1844 

.8663 

.1978 

.8166 

.2120 

.7700 

36 

25 

1.1596 

1.9752 

1.1718 

1.9184 

1.1846 

1.8654 

1.1981 

1.8158 

1.2122 

1.7693 

35 

26 

.1598 

.9742 

.1720 

.9175 

.1848 

.8646 

.1983 

.8150 

.2124 

.7685 

34 

27 

.1600 

.9732 

.1722 

.9166 

.1850 

.8637 

.1985 

.8142 

.2127 

.7678 

33 

28 

.1602 

.9722 

.1724 

.9157 

.1852 

.8629 

.1987 

.8134 

.2129 

.7670 

32 

29 

.1604 

.9713 

.1726 

.9148 

.1855 

.8620 

.1990 

.8126 

.2132 

.7663 

31 

30 

1.1606 

1.9703 

1.1728 

1.9139 

1.1857 

1.8612 

1.1992 

1.8118 

1.2134 

1.7655 

30 1 

31 

.1608 

.9693 

.1730 

.9130 

.1859 

.8603 

.1994 

.8110 

.2136 

.7648 

29 

32 

.1610 

.9684 

.1732 

.9121 

.1861 

.8595 

.1997 

.8102 

.2139 

.7640 

28 1 

33 

.1612 

.9674 

.1735 

.9112 

.1863 

.8586 

.1999 

.8094 

.2141 

.7633 

27 

34 

.1614 

.9664 

.1737 

.9103 

.1866 

.8578 

.2001 

.8086 

.2144 

.7625 

26 1 

35 

1.1616 

1.9654 

1.1739 

1.9094 

1.1868 

1.8569 

1.2004 

1.8078 

1.2146 

1.7618 

25 1 

36 

.1618 

.9645 

.1741 

.9084 

.1870 

.8561 

.2006 

.8070 

.2149 

.7610 

24 I 

37 

.1620 

.9635 

.1743 

.9075 

.1872 

.8552 

.2008 

.8062 

.2151 

.7603 

23 

38 

.1622 

.9625 

.1745 

.9066 

.1875 

.8544 

.2011 

.8055 

.2154 

.7596 

22 

39 

.1624 

.9616 

.1747 

.9057 

.1877 

.8535 

.2013 

.8047 

.2156 

.7588 

21 1 

40 

1.1626 

1.9606 

1.1749 

1.9048 

1.1879 

1.8527 

1.2015 

1.8039 

1.2158 

1.7581 

20 

41 

.1628 

.9597 

.1751 

.9039 

.1881 

.8519 

.2018 

.8031 

.2161 

.7573 

19 

42 

.1630 

.9587 

.1753 

.9031 

.1883 

.8510 

.2020 

.8023 

.2163 

.7566 

18 

43 

.1632 

.9577 

.1756 

.9022 

.1886 

.8502 

.2022 

.8015 

.2166 

.7559 

17 

44 

.1634 

.956S 

.1758 

.9013 

.1888 

.8494 

.2025 

.8007 

.2168 

.7551 

16 

45 

1.1636 

1.955S 

1.1760 

1.9004 

1.1890 

1.8485 

1.2027 

1.8000 

1.2171 

1.7544 

15 

46 

.1638 

.9549 

.1762 

.8995 

.1892 

.8477 

.2029 

.7992 

.2173 

.7537 

14 

47 

.1640 

.9539 

.1764 

.8986 

.1895 

.8468 

.2032 

.7984 

.2176 

.7529 

13 

48 

.1642 

.9530 

.1766 

.8977 

.1897 

.8460 

.2034 

.7976 

.2178 

.7522 

12 

49 

.1644 

.9520 

.1768 

.8968 

.1899 

.8452 

.2036 

.7968 

.2181 

.7515 

11 

50 

1.1646 

1.9511 

1.1770 

1.8959 

1.1901 

1.8443 

1.2039 

1.7960 

1.2183 

1.7507 

10 

51 

.1648 

.9501 

.1773 

.8950 

.1903 

.8435 

.2041 

.7953 

.2185 

.7500 

9 

52 

.1650 

.9492 

.1775 

.8941 

.1906 

.8427 

.2043 

.7945 

.2188 

.7493 

8~1 

53 

.1652 

.9482 

.1777 

.8933 

.1908 

.8419 

.2046 

.7937 

.2190 

.7485 

7| 

54 

.1654 

.9473 

.1779 

.8924 

.1910 

.8410 

.2048 

.7929 

.2193 

.7478 

6| 

55 

1.1656 

1.9463 

1.1781 

1,8915 

1.1912 

1.8402 

1.2050 

1.7922 

1.2195 

1.7471 

5 

56 

.1658 

.9454 

.1783 

.8906 

.1915 

.8394 

.2053 

.7914 

.2198 

.7463 

4 

57 

.1660 

.9444 

.1785 

.8897 

.1917 

.8385 

.2055 

.7906 

.2200 

.7456 

3| 

58 

.1662 

.9435 

.1788 

.8888 

.1919 

.8377 

.2057 

.7898 

.2203 

.7449 

2 \ 

59 

.1664 

.9425 

.1790 

.8880 

.1921 

.8369 

.2060 

.7891 

.2205 

.7442 

ll 

60 

1.1666 

1.9416 

1.1792 

1.8871 

1.1924 

1.8361 

1.2062 

1.7883 

1.2208 

1.7434 

0| 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ I 


59° 

58 

o 

57 

o 

56 c 

— 

3 

55 c 


<- 


[ 112 ] 























































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 


->■ 

35° 

36° 

37° 

38° 

39° 


/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.2208 

1.7434 

1.2361 

1.7013 

1.2521 

1.6616 

1.2690 

1.6243 

1.2868 

1.5890 

60 

1 

.2210 

.7427 

.2363 

.7006 

.2524 

.6610 

.2693 

.6237 

.2871 

.5884 

59 

2 

.2213 

.7420 

.2366 

.6999 

.2527 

.6604 

.2696 

.6231 

.2874 

.5879 

58 

3 

.2215 

.7413 

.2369 

.6993 

.2530 

.6597 

.2699 

.6225 

.2877 

.5873 

57 

4 

.2218 

.7400 

.2371 

.6986 

.2532 

.6591 

.2702 

.6219 

.2880 

.5867 

56 

5 

1.2220 

1.7398 

1.2374 

1.6979 

1.2535 

1.6584 

1.2705 

1.6213 

1.2883 

1.5862 

55 

6 

.2223 

.7391 

.2376 

.6972 

.2538 

.6578 

.2708 

.6207 

.2886 

.5856 

54 

7 

.2225 

.7384 

.2379 

.6966 

.2541 

.6572 

.2710 

.6201 

.2889 

.5850 

53 

8 

.2228 

.7377 

.2382 

.6959 

.2543 

.6565 

.2713 

.6195 

.2892 

.5845 

52 

9 

.2230 

.7370 

.2384 

.6952 

.2546 

.6559 

.2716 

.6189 

.2895 

.6839 

51 

10 

1.2233 

1.7362 

1.2387 

1.6945 

1.2549 

1.6553 

1.2719 

1.6183 

1.2898 

1.5833 

50 

11 

.2235 

.7355 

.2390 

.6939 

.2552 

.6546 

.2722 

.6177 

.2901 

.5828 

49 

12 

.2238 

.7348 

.2392 

.6932 

.2554 

.6540 

.2725 

.6171 

.2904 

.5822 

48 

13 

.2240 

.7341 

.2395 

.6925 

.2557 

.6534 

.2728 

.6165 

.2907 

.5816 

47 

14 

.2243 

.7334 

.2397 

.6918 

.2560 

.6527 

.2731 

.6159 

.2910 

.5811 

46 

15 

1.2245 

1.7327 

1.2400 

1.6912 

1.2563 

1.6521 

1.2734 

1.6153 

1.2913 

1.5805 

45 

16 

.2248 

.7320 

.2403 

.6905 

.2566 

.6515 

.2737 

.6147 

.2916 

.5800 

44 

17 

.2250 

.7312 

.2405 

.6898 

.2568 

.6508 

.2740 

.6141 

.2919 

.5794 

43 

18 

.2253 

.7305 

.2408 

.6892 

.2571 

.6502 

.2742 

.6135 

.2923 

.5788 

42 

19 

.2255 

.7298 

.2411 

.6885 

.2574 

.6496 

.2745 

.6129 

.2926 

.5783 

41 

20 

1.2258 

1.7291 

1.2413 

1.6878 

1.2577 

1.6489 

1.2748 

1.6123 

1.2929 

1.5777 

40 

21 

.2260 

.7284 

.2416 

.6871 

.2579 

.6483 

.2751 

.6117 

.2932 

.5771 

39 

22 

.2263 

.7277 

.2419 

.6865 

.2582 

.6477 

.2754 

.6111 

.2935 

.5766 

38 

23 

.2265 

.7270 

.2421 

.6858 

.2585 

.6471 

.2757 

.6105 

.2938 

.5760 

37 

24 

.2268 

.7263 

.2424 

.6852 

.2588 

.6464 

.2760 

.6099 

.2941 

.5755 

36 

25 

1.2271 

1.7256 

1.2427 

1.6845 

1.2591 

1.6458 

1.2763 

1.6093 

1.2944 

1.5749 

35 

26 

.2273 

.7249 

.2429 

.6838 

.2593 

.6452 

.2766 

.6087 

.2947 

.5744 

34 

27 

.2276 

.7242 

.2432 

.6832 

.2596 

.6446 

.2769 

.6082 

.2950 

.5738 

33 

28 

.2278 

.7235 

.2435 

.6825 

.2599 

.6439 

.2772 

.6076 

.2953 

.5732 

32 

29 

.2281 

.7228 

.2437 

.6818 

.2602 

.6433 

.2775 

.6070 

.2957 

.5727 

31 

30 

1.2283 

1.7221 

1.2440 

1.6812 

1.2605 

1.6427 

1.2778 

1.6064 

1.2960 

1.5721 

30 

31 

.2286 

.7213 

.2443 

.6805 

.2608 

.6421 

.2781 

.6058 

.2963 

.5716 

29 

32 

.2288 

.7206 

.2445 

.6799 

.2610 

.6414 

.2784 

.6052 

.2966 

.5710 

28 

33 

.2291 

.7199 

.2448 

.6792 

.2613 

.6408 

.2787 

.6046 

.2969 

.5705 

27 

34 

.2293 

.7192 

.2451 

.6785 

.2616 

.6402 

.2790 

.6040 

.2972 

.5699 

26 

35 

1.2296 

1.7185 

1.2453 

1.6779 

1.2619 

1.6396 

1.2793 

1.6035 

1.2975 

1.5694 

25 

36 

.2299 

.7179 

.2456 

.6772 

.2622 

.6390 

.2796 

.6029 

.2978 

.5688 

24 

37 

.2301 

.7172 

.2459 

.6766 

.2624 

.6383 

.2799 

.6023 

.2981 

.5683 

23 

38 

.2304 

.7165 

.2462 

.6759 

.2627 

.6377 

.2802 

.6017 

.2985 

.5677 

22 

39 

.2306 

.7158 

.2464 

.6753 

.2630 

.6371 

.2804 

.6011 

.2988 

.5672 

21 

40 

1.2309 

1.7151 

1.2467 

1.6746 

1.2633 

1.6365 

1.2807 

1.6005 

1.2991 

1.5666 

20 

41 

.2311 

.7144 

.2470 

.6739 

.2636 

.6359 

.2810 

.6000 

.2994 

.5661 

19 

42 

.2314 

.7137 

.2472 

.6733 

.2639 

.6353 

.2813 

.5994 

.2997 

.5655 

18 

43 

.2317 

.7130 

.2475 

.6726 

.2641 

.6346 

.2816 

.5988 

.3000 

.5650 

17 

44 

.2319 

.7123 

.2478 

.6720 

.2644 

.6340 

.2819 

.5982 

.3003 

.5644 

16 

45 

1.2322 

1.7116 

1.2480 

1.6713 

1.2647 

1.6334 

1.2822 

1.5976 

1.3007 

1.5639 

15 

46 

.2324 

.7109 

.2483 

.6707 

.2650 

.6328 

.2825 

.5971 

.3010 

.5633 

14 

47 

.2327 

.7102 

.2486 

.6700 

.2653 

.6322 

.2828 

.5965 

.3013 

.5628 

13 

48 

.2329 

.7095 

.2489 

.6694 

.2656 

.6316 

.2831 

.5959 

.3016 

.5622 

12 

49 

.2332 

.7088 

.2491 

.6687 

.2659 

.6310 

.2834 

.5953 

.3019 

.5617 

11 

50 

1.2335 

1.7081 

1.2494 

1.6681 

1.2661 

1.6303 

1.2837 

1.5948 

1.3022 

1.5611 

10 

51 

.2337 

.7075 

.2497 

.6674 

.2664 

.6297 

.2840 

.5942 

.3026 

.5606 

9 

52 

.2340 

.7068 

.2499 

.6668 

.2667 

.6291 

.2843 

.5936 

.3029 

.5601 

8 

53 

.2342 

.7061 

.2502 

.6661 

.2670 

.6285 

.2846 

.5930 

.3032 

.5595 

7 

54 

.2345 

.7054 

.2505 

.6655 

.2673 

.6279 

.2849 

.5925 

.3035 

.5590 

6 

55 

1.2348 

1.7047 

1.2508 

1.6649 

1.2676 

1.6273 

1.2852 

1.5919 

1.3038 

1.5584 

5 

56 

.2350 

.7040 

.2510 

.6642 

.2679 

.6267 

.2855 

.5913 

.3041 

.5579 

4 

57 

.2353 

.7033 

.2513 

.6636 

.2682 

.6261 

.2859 

.5907 

.3045 

.5573 

3 

58 

.2355 

.7027 

.2516 

.6629 

.2684 

.6255 

.2862 

.5902 

.3048 

.5568 

2 

59 

.2358 

.7020 

.2519 

.6623 

.2687 

.6249 

.2865 

.5896 

.3051 

.5563 

1 

60 

1.2361 

1.7013 

1.2521 

1.6616 

1.2690 

1.6243 

1.2868 

1.5890 

1.3054 

1.5557 

0 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 


54° 

53° 

52° 

51 

o 

50 

O 



[ 113 ] 










































































































XI. FIVE-PLACE VALUES: SECANT AND COSECANT 



o 

O 

41° 

42° 

43° 

44° 


/ 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 


0 

1.3054 

1.5557 

1.3250 

1.5243 

1.3456 

1.4945 

1.3673 

1.4663 

1.3902 

1.4396 

60 

1 

.3057 

.5552 

.3253 

.5237 

.3460 

.4940 

.3677 

.4658 

.3906 

.4391 

59 

2 

.3060 

.5546 

.3257 

.5232 

.3463 

.4935 

.3681 

.4654 

.3909 

.4387 

58 

3 

.3064 

.5541 

.3260 

.5227 

.3467 

.4930 

.3684 

.4649 

.3913 

.4383 

57 

4 

.3067 

.5536 

.3264 

.5222 

.3470 

.4925 

.3688 

.4645 

.3917 

.4378 

56 

5 

1.3070 

1.5530 

1.3267 

1.5217 

1.3474 

1.4921 

1.3692 

1.4640 

1.3921 

1.4374 

55 

6 

.3073 

.5525 

.3270 

.5212 

.3478 

.4916 

.3696 

.4635 

.3925 

.4370 

54 

7 

.3076 

.5520 

.3274 

.5207 

.3481 

.4911 

.3699 

.4631 

.3929 

.4365 

53 

8 

.3080 

.5514 

.3277 

.5202 

.3485 

.4906 

.3703 

.4626 

.3933 

.4361 

52 

9 

.3083 

.5509 

.3280 

.5197 

.3488 

.4901 

.3707 

.4622 

.3937 

.4357 

51 

10 

1.3086 

1.5504 

1.3284 

1.5192 

1.3492 

1.4897 

1.3711 

1.4617 

1.3941 

1.4352 

50 

11 

.3089 

.5498 

.3287 

.5187 

.3495 

.4892 

.3714 

.4613 

.3945 

.4348 

49 

12 

.3093 

.5493 

.3291 

.5182 

.3499 

.4887 

.3718 

.4608 

.3949 

.4344 

48 

13 

.3096 

.5488 

.3294 

.5177 

.3502 

.4882 

.3722 

.4604 

.3953 

.4340 

47 

14 

.3099 

.5482 

.3297 

.5172 

.3506 

.4878 

.3726 

.4599 

.3957 

.4335 

46 

15 

1.3102 

1.5477 

1.3301 

1.5167 

1.3510 

1.4873 

1.3729 

1.4595 

1.3961 

1.4331 

45 

16 

.3105 

.5472 

.3304 

.5162 

.3513 

.4868 

.3733 

.4590 

.3965 

.4327 

44 

17 

.3109 

.5466 

.3307 

.5156 

.3517 

.4S63 

.3737 

.4586 

.3969 

.4322 

43 

18 

.3112 

.5461 

.3311 

.5151 

.3520 

.4859 

.3741 

.4581 

.3972 

.4318 

42 

19 

.3115 

.5456 

.3314 

.5146 

.3524 

.4854 

.3744 

.4577 

.3976 

.4314 

41 

20 

1.3118 

1.5450 

1.3318 

1.5141 

1.3527 

1.4849 

1.3748 

1.4572 

1.3980 

1.4310 

40 

21 

.3122 

.5445 

.3321 

.5136 

.3531 

.4844 

.3752 

.4568 

.3984 

.4305 

39 

22 

.3125 

.5440 

.3325 

.5131 

.3535 

.4840 

.3756 

.4563 

.3988 

.4301 

38 

23 

.3128 

.5435 

.3328 

.5126 

.3538 

.4835 

.3759 

.4559 

.3992 

.4297 

37 

24 

.3131 

.5429 

.3331 

.5121 

.3542 

.4830 

.3763 

.4554 

.3996 

.4293 

36 

25 

1.3135 

1.5424 

1.3335 

1.5116 

1.3545 

1.4825 

1.3767 

1.4550 

1.4000 

1.4288 

35 

26 

.3138 

.5419 

.3338 

.5111 

.3549 

.4821 

.3771 

.4545 

.4004 

.4284 

34 

27 

.3141 

.5413 

.3342 

.5107 

.3553 

.4816 

.3775 

.4541 

.4008 

.4280 

33 

28 

.3144 

.5408 

.3345 

.5102 

.3556 

.4811 

.3778 

.4536 

.4012 

.4276 

32 

29 

.3148 

.5403 

.3348 

.5097 

.3560 

.4807 

.3782 

.4532 

.4016 

.4271 

31 

30 

1.3151 

1.5398 

1.3352 

1.5092 

1.3563 

1.4802 

1.3786 

1.4527 

1.4020 

1.4267 

30 

31 

.3154 

.5392 

.3355 

.5087 

.3567 

.4797 

.3790 

.4523 

.4024 

.4263 

29 

32 

.3157 

.5387 

.3359 

.5082 

.3571 

.4792 

.3794 

.4518 

.4028 

.4259 

28 

33 

.3161 

.5382 

.3362 

.5077 

.3574 

.4788 

.3797 

.4514 

.4032 

.4255 

27 

34 

.3164 

.5377 

.3366 

.5072 

.3578 

.4783 

.3801 

.4510 

.4036 

.4250 

26 

35 

1.3167 

1.5372 

1.3369 

1.5067 

1.3582 

1.4778 

1.3805 

1.4505 

1.4040 

1.4246 

25 

36 

.3171 

.5366 

.3373 

.5062 

.3585 

.4774 

.3809 

.4501 

.4044 

.4242 

24 

37 

.3174 

.5361 

.3376 

.5057 

.3589 

.4769 

.3813 

.4496 

.4048 

.4238 

23 

38 

.3177 

.5356 

.3380 

.5052 

.3592 

.4764 

.3817 

.4492 

.4052 

.4234 

22 

39 

.3180 

.5351 

.3383 

.5047 

.3596 

.4760 

.3820 

.4487 

.4057 

.4229 

21 

40 

1.3184 

1.5345 

1.3386 

1.5042 

1.3600 

1.4755 

1.3824 

1.4483 

1.4061 

1.4225 

20 

41 

.3187 

.5340 

.3390 

.5037 

.3603 

.4750 

.3828 

.4479 

.4065 

.4221 

19 

42 

.3190 

.5335 

.3393 

.5032 

.3607 

.4746 

.3832 

.4474 

.4069 

.4217 

18 

43 

.3194 

.5330 

.3397 

.5027 

.3611 

.4741 

.3836 

.4470 

.4073 

.4213 

17 

44 

.3197 

.5325 

.3400 

.5023 

.3614 

.4737 

.3840 

.4465 

.4077 

.4208 

16 

45 

1.3200 

1.5320 

1.3404 

1.5018 

1.3618 

1.4732 

1.3843 

1.4461 

1.4081 

1.4204 

15 

46 

.3203 

.5314 

.3407 

.5013 

.3622 

.4727 

.3847 

.4457 

.4085 

.4200 

14 

47 

.3207 

.5309 

.3411 

.5008 

.3625 

.4723 

.3851 

.4452 

.4089 

.4196 

13 

48 

.3210 

.5304 

.3414 

.5003 

.3629 

.4718 

.3855 

.4448 

.4093 

.4192 

12 

49 

.3213 

.5299 

.3418 

.4998 

.3633 

.4713 

.3859 

.4443 

.4097 

.4188 

11 

50 

1.3217 

1.5294 

1.3421 

1.4993 

1.3636 

1.4709 

1.3863 

1.4439 

1.4101 

1.4183 

10 

51 

.3220 

.5289 

.3425 

.4988 

.3640 

.4704 

.3867 

.4435 

.4105 

.4179 

9 

52 

.3223 

.5283 

.3428 

.4984 

.3644 

.4700 

.3871 

.4430 

.4109 

.4175 

8 

53 

.3227 

.5278 

.3432 

.4979 

.3647 

.4695 

.3874 

.4426 

.4113 

.4171 

7 

54 

.3230 

.5273 

.3435 

.4974 

.3651 

.4690 

.3878 

.4422 

.4118 

.4167 

6 

55 

1.3233 

1.5268 

1.3439 

1.4969 

1.3655 

1.4686 

1.3882 

1.4417 

1.4122 

1.4163 

5 

56 

.3237 

.5263 

.3442 

.4964 

.3658 

.4681 

.3886 

.4413 

.4126 

.4159 

4 

57 

.3240 

.5258 

.3446 

.4959 

.3662 

.4677 

.3890 

.4409 

.4130 

.4154 

3 

58 

.3243 

.5253 

.3449 

.4954 

.3666 

.4672 

.3894 

.4404 

.4134 

.4150 

2 

59 

.3247 

.5248 

.3453 

.4950 

.3670 

.4667 

.3898 

.4400 

.4138 

.4146 

1 

60 

1.3250 

1.5243 

1.3456 

1.4945 

1.3673 

1.4663 

1.3902 

1.4396 

1.4142 

1.4142 

0 


Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

Csc 

Sec 

/ 


49° 

oo 

o 

47° 

46° 

45 

o 



[ 114 ] 





































































































XII. SQUARES AND SQUARE ROOTS 


N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

V ION 

1.00 

1.01 

1.02 

1.03 

1.04 

1.05 

1.06 

1.07 

1.08 

1.09 

1.10 

1.11 

1.12 

1.13 

1.14 

1.15 

1.16 

1.17 

1.18 

1.19 

1.20 

1.21 

1.22 

1.23 

1.24 

1.25 

1.26 

1.27 

1.28 

1.29 

1.30 

1.0000 

1.0201 

1.0404 

1.0609 

1.0816 

1.1025 

1.1236 

1.1449 

1.1664 

1.1881 

1.2100 

1.00000 

1.00499 

1.00995 

1.01489 

1.01980 

1.02470 

1.02956 

1.03441 

1.03923 

1.04403 

1.048S1 

3.16228 

3.17805 

3.19374 

3.20936 

3.22490 

3.24037 

3.25576 

3.27109 

3.28634 

3.30151 

3.31662 

1.60 

1.61 

1.62 

1.63 

1.64 

1.65 

1.66 

1.67 

1.68 

1.69 

1.70 

1.71 

1.72 

1.73 

1.74 

1.75 

1.76 

1.77 

1.78 

1.79 

1.80 

1.81 

1.82 

1.83 

1.84 

1.85 

1.86 

1.87 

1.88 

1.89 

1.90 

2.5600 

2.5921 

2.6244 

2.6569 

2.6896 

2.7225 

2.7556 

2.7889 

2.8224 

2.8561 

2.8900 

1.26491 

1.26886 

1.27279 

1.27671 

1.28062 

1.28452 

1.28841 

1.29228 

1.29615 

1.30000 

1.30384 

4.00000 

4.01248 

4.02492 

4.03733 

4.04969 

4.06202 

4.07431 

4.08656 

4.09878 

4.11096 

4.12311 

2.20 

2.21 

2.22 

2.23 

2.24 

2.25 

2.26 

2.27 

2.28 

2.29 

2.30 

2.31 

2.32 

2.33 

2.34 

2.35 

2.36 

2.37 

2.38 

2.39 

2.40 

2.41 

2.42 

2.43 

2.44 

2.45 

2.46 

2.47 

2.48 

2.49 

2.50 

4.8400 

4.8841 

4.9284 

4.9729 

5.0176 

5.0625 

5.1076 

5.1529 

5.1984 

5.2441 

5.2900 

1.48324 

1.48661 

1.48997 

1.49332 

1.49666 

1.50000 

1.50333 

1.50665 

1.50997 

1.51327 

1.51658 

4.69042 

4.70106 

4.71169 

4.72229 

4.73286 

4.74342 

4.75395 

4.76445 

4.77493 

4.78539 

4.79583 

1.2321 

1.2544 

1.2769 

1.2996 

1.3225 

1.3456 

1.3689 

1.3924 

1.4161 

1.4400 

1.05357 

1.05830 

1.06301 

1.06771 

1.07238 

1.07703 

1.08167 

1.08628 

1.09087 

1.09545 

3.33167 

3.34664 

3.36155 

3.37639 

3.39116 

3.40588 

3.42053 

3.43511 

3.44964 

3.46410 

2,9241 

2.9584 

2.9929 

3.0276 

3.0625 

3.0976 

3.1329 

3.1684 

3.2041 

3.2400 

1.30767 

1.31149 

1.31529 

1.31909 

1.32288 

1.32665 

1.33041 

1.33417 

1.33791 

1.34164 

4.13521 

4.14729 

4.15933 

4.17133 

4.18330 

4.19524 

4.20714 

4.21900 

4.23084 

4.24264 

5.3361 

5.3824 

5.4289 

5.4756 

5.5225 

5.5696 

5.6169 

5.6644 

5.7121 

5.7600 

1.51987 

1.52315 

1.52643 

1.52971 

1.53297 

1.53623 

1.53948 

1.54272 

1.54596 

1.54919 

4.80625 

4.81664 

4.82701 

4.83735 

4.84768 

4.85798 

4.86826 

4.87852 

4.88876 

4.89898 

1.4641 

1.4884 

1.5129 

1.5376 

1.5625 

1.5876 

1.6129 

1.6384 

1.6641 

1.6900 

1.10000 

1.10454 

1.10905 

1.11355 

1.11803 

1.12250 

1.12694 

1.13137 

1.13578 

1.14018 

3.47851 

3.49285 

3.50714 

3.52136 

3.53553 

3.54965 

3.56371 

3.57771 

3.59166 

3.60555 

3.2761 

3.3124 

3.3489 

3.3856 

3.4225 

3.4596 

3.4969 

3.5344 

3.5721 

3.6100 

1.34536 

1.34907 

1.35277 

1.35647 

1.36015 

1.36382 

1.36748 

1.37113 

1.37477 

1.37840 

4.25441 

4.26615 

4.27785 

4.28952 

4.30116 

4.31277 

4.32435 

4.33590 

4.34741 

4.35890 

5.8081 

5.8564 

5.9049 

5.9536 

6.0025 

6.0516 

6.1009 

6.1504 

6.2001 

6.2500 

1.55242 

1.55563 

1.55885 

1.56205 

1.56525 

1.56844 

1.57162 

1.57480 

1.57797 

1.58114 

4.90918 

4.91935 

4.92950 

4.93964 

4.94975 

4.95984 

4.96991 

4.97996 

4.98999 

5.00000 

1.31 

1.7161 

1.14455 

3.61939 

1.91 

3.6481 

1.38203 

4.37035 

2.51 

6.3001 

1.58430 

5.00999 

1.32 

1.7424 

1.14891 

3.63318 

1.92 

3.6864 

1.38564 

4.38178 

2.52 

6.3504 

1.58745 

5 01996 

1.33 

1.7689 

1.15326 

3.64692 

1.93 

3.7249 

1.38924 

4.39318 

2.53 

6.4009 

1.59060 

5 02991 

1.34 

1.7956 

1.15758 

3.66060 

1.94 

3.7636 

1.39284 

4.40454 

2.54 

6.4516 

1.59374 

5 03984 

1.35 

1.8225 

1.16190 

3.67423 

1.95 

3.8025 

1.39642 

4.41588 

2.55 

6.5025 

1.59687 

5.04975 

1.36 

1.8496 

1.16619 

3.68782 

1.96 

3.8416 

1.40000 

4.42719 

2.56 

6.5536 

1.60000 

5.05964 

1.37 

1.8769 

1.17047 

3.70135 

1.97 

3.8S09 

1.40357 

4.43847 

2.57 

6.6049 

1.60312 

5.06952 

1.38 

1.9044 

1.17473 

3.71484 

1.98 

3.9204 

1.40712 

4.44972 

2.58 

6.6564 

1.60624 

5.07937 

1.39 

1.9321 

1.17898 

3.72827 

1.99 

3.9601 

1.41067 

4.46094 

2.59 

6.7081 

1.60935 

5.08920 

1.40 

1.9600 

1.18322 

3.74166 

2.00 

4.0000 

1.41421 

4.47214 

2.60 

6.7600 

1.61245 

5.09902 

1.41 

1.9881 

1.18743 

3.75500 

2.01 

4.0401 

1.41774 

4.48330 

2.61 

6.8121 

1.61555 

5.10882 

1.42 

2.0164 

1.19164 

3.76829 

2.02 

4.0804 

1.42127 

4.49444 

2.62 

6.8644 

1.61864 

5.11859 

1.43 

2.0449 

1.19583 

3.78153 

2.03 

4.1209 

1.42478 

4.50555 

2.63 

6.9169 

1.62173 

5.12835 

1.44 

2.0736 

1.20000 

3.79473 

2.04 

4.1616 

1.42829 

4.51664 

2.64 

6.9696 

1.62481 

5.13809 

1.45 

2.1025 

1.20416 

3.80789 

2.05 

4.2025 

1.43178 

4.52769 

2.65 

7.0225 

1.62788 

5.14782 

1.46 

2.1316 

1.20830 

3.82099 

2.06 

4.2436 

1.43527 

4.53872 

2.66 

7.0756 

1.63095 

5.15752 

1.47 

2.1609 

1.21244 

3.83406 

2.07 

4.2849 

1.43875 

4.54973 

2.67 

7.1289 

1.63401 

5.16720 

1.48 

2.1904 

1.21655 

3.84708 

2.08 

4.3264 

1.44222 

4.56070 

2.68 

7.1824 

1.63707 

5.17687 

1.49 

2.2201 

1.22066 

3.86005 

2.09 

4.3681 

1.44568 

4.57165 

2.69 

7.2361 

1.64012 

5.1S652 

1.50 

2.2500 

1.22474 

3.87298 

2.10 

4.4100 

1.44914 

4.58258 

2.70 

7.2900 

1.64317 

5.19615 

1.51 

2.2801 

1.22882 

3.88587 

2.11 

4.4521 

1.45258 

4.59347 

2.71 

7.3441 

1.64621 

5.20577 

1.52 

2.3104 

1.23288 

3.89872 

2.12 

4.4944 

1.45602 

4.60435 

2.72 

7.3984 

1.64924 

5.21536 

1.53 

2.3409 

1.23693 

3.91152 

2.13 

4.5369 

1.45945 

4.61519 

2.73 

7.4529 

1.65227 

5.22494 

1.54 

2.3716 

1.24097 

3.92428 

2.14 

4.5796 

1.46287 

4.62601 

2.74 

7.5076 

1.65529 

5.23450 

1.55 

2.4025 

1.24499 

3.93700 

2.15 

4.6225 

1.46629 

4.63681 

2.75 

7.5625 

1.65831 

5.24404 

1.56 

2.4336 

1.24900 

3.94968 

2.16 

4.6656 

1.46969 

4.64758 

2.76 

7.6176 

1.66132 

5.25357 

1.57 

2.4649 

1.25300 

3.96232 

2.17 

4.7089 

1.47309 

4.65833 

2.77 

7.6729 . 

1.66433 

5.26308 

1.58 

2.4964 

1.25698 

3.97492 

2.18 

4.7524 

1.47648 

4.66905 

2.78 

7.7284 

1.66733 

5.27257 

1.59 

2.5281 

1.26095 

3.98748 

2.19 

4.7961 

1.47986 

4.67974 

2.79 

7.7841 

1.67033 

5.28205 

1.60 

2.5600 

1.26491 

4.00000 

2.20 

4.8400 

1.48324 

4.69042 

2.80 

7.8400 

1.67332 

5.29150 

UL- 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

V ION 

N 

N 2 | 

Vn 

Vl ON 


[H5] 


























































































































XII. SQUARES AND SQUARE ROOTS 


N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

VlON 

2.80 

7.8400 

1.67332 

5.29150 

3.40 

11.5600 

1.84391 

5.83095 

4.00 

16.0000 

2.00000 

6.32456 

2.81 

7.8961 

1.67631 

5.30094 

3.41 

11.6281 

1.84662 

5.83952 

4.01 

16.0801 

2.00250 

6.33246 

2.82 

7.9524 

1.67929 

5.31037 

3.42 

11.6964 

1.84932 

5.84808 

4.02 

16.1604 

2.00499 

6.34035 

2.83 

8.0089 

1.68226 

5.31977 

3.43 

11.7649 

1.85203 

5.85662 

4.03 

16.2409 

2.00749 

6.34823 

2.84 

8.0656 

1.68523 

5.32917 

3.44 

11.8336 

1.85472 

5.86515 

4.04 

16.3216 

2.00998 

6.35610 

2.85 

8.1225 

1.68819 

5.33854 

3.45 

11.9025 

1.85742 

5.87367 

4.05 

16.4025 

2.01246 

6.36396 

2.86 

8.1796 

1.69115 

5.34790 

3.46 

11.9716 

1.86011 

5.88218 

4.06 

16.4836 

2.01494 

6.37181 

2.87 

8.2369 

1.69411 

5.35724 

3.47 

12.0409 

1.86279 

5.89067 

4.07 

16.5649 

2.01742 

6.37966 

2.88 

8.2944 

1.69706 

5.36656 

3.48 

12.1104 

1.86548 

5.89915 

4.08 

16.6464 

2.01990 

6.38749 

2.89 

8.3521 

1.70000 

5.37587 

3.49 

12.1801 

1.86815 

5.90762 

4.09 

16.7281 

2.02237 

6.39531 

2.90 

8.4100 

1.70294 

5.38516 

3.50 

12.2500 

1.87083 

5.91608 

4.10 

16.8100 

2.02485 

6.40312 

2.91 

8.4681 

1.70587 

5.39444 

3.51 

12.3201 

1.87350 

5.92453 

4.11 

16.8921 

2.02731 

6.41093 

2.92 

8.5264 

1.70880 

5.40370 

3.52 

12.3904 

1.87617 

5.93296 

4.12 

16.9744 

2.02978 

6.41872 

2.93 

8.5849 

1.71172 

5.41295 

3.53 

12.4609 

1.87883 

5.94138 

4.13 

17.0569 

2.03224 

6.42651 

2.94 

8.6436 

1.71464 

5.42218 

3.54 

12.5316 

1.88149 

5.94979 

4.14 

17.1396 

2.03470 

6.43428 

2.95 

8.7025 

1.71756 

5.43139 

3.55 

12.6025 

1.88414 

5.95819 

4.15 

17.2225 

2.03715 

6.44205 

2.96 

8.7616 

1.72047 

5.44059 

3.56 

12.6736 

1.88680 

5.96657 

4.16 

17.3056 

2,03961 

6.44981 

2.97 

8.8209 

1.72337 

5.44977 

3.57 

12.7449 

1.88944 

5.97495 

4.17 

17.3889 

2.04206 

6.45755 

2.98 

8.8804 

1.72627 

5.45894 

3.58 

12.8164 

1.89209 

5.98331 

4.18 

17.4724 

2.04450 

6.46529 

2.99 

8.9401 

1.72916 

5.46809 

3.59 

12.8881 

1.89473 

5.99166 

4.19 

17.5561 

2.04695 

6.47302 

3.00 

9.0000 

1.73205 

5.47723 

3.60 

12.9600 

1.89737 

6.00000 

4.20 

17.6400 

2.04939 

6.48074 

3.01 

9.0601 

1.73494 

5.48635 

3.61 

13.0321 

1.90000 

6.00833 

4.21 

17.7241 

2.05183 

6.48845 

3.02 

9.1204 

1.73781 

5.49545 

3.62 

13.1044 

1.90263 

6.01664 

4.22 

17.8084 

2.05426 

6.49615 

3.03 

9.1809 

1.74069 

5.50454 

3.63 

13.1769 

1.90526 

6.02495 

4.23 

17.8929 

2.05670 

6.50384 

3.04 

9.2416 

1.74356 

5.51362 

3.64 

13.2496 

1.90788 

6.03324 

4.24 

17.9776 

2.05913 

6.51153 

3.05 

9.3025 

1.74642 

5.52268 

3.65 

13.3225 

1.91050 

6.04152 

4.25 

18.0625 

2.06155 

6.51920 

3.06 

9.3636 

1.74929 

5.53173 

3.66 

13.3956 

1.91311 

6.04979 

4.26 

18.1476 

2.06398 

6.52687 

3.07 

9.4249 

1.75214 

5.54076 

3.67 

13.4689 

1.91572 

6.05805 

4.27 

18.2329 

2.06640 

6.53452 

3.08 

9.4864 

1.75499 

5.54977 

3.68 

13.5424 

1.91833 

6.06630 

4.28 

18.3184 

2.06882 

6.54217 

3.09 

9.5481 

1.75784 

5.55878 

3.69 

13.6161 

1.92094 

6.07454 

4.29 

18.4041 

2.07123 

6.54981 

3.10 

9.6100 

1.76068 

5.56776 

3.70 

13.6900 

1.92354 

6.08276 

4.30 

18.4900 

2.07364 

6.55744 

3.11 

9.6721 

1.76352 

5.57674 

3.71 

13.7641 

1.92614 

6.09098 

4.31 

18.5761 

2.07605 

6.56506 

3.12 

9.7344 

1.76635 

5.58570 

3.72 

13.8384 

1.92873 

6.09918 

4.32 

18.6624 

2.07846 

6.57267 

3.13 

9.7969 

1.76918 

5.59464 

3.73 

13.9129 

1.93132 

6.10737 

4.33 

18.7489 

2.08087 

6.58027 

3.14 

9.8596 

1.77200 

5.60357 

3.74 

13.9876 

1.93391 

6.11555 

4.34 

18.8356 

2.08327 

6.58787 

3.15 

9.9225 

1.77482 

5.61249 

3.75 

14.0625 

1.93649 

6.12372 

4.35 

18.9225 

2.08567 

6.59545 

3.16 

9.9856 

1.77764 

5.62139 

3.76 

14.1376 

1.93907 

6.13188 

4.36 

19.0096 

2.08806 

6.60303 

3.17 

10.0489 

1.78045 

5.63028 

3.77 

14.2129 

1.94165 

6.14003 

4.37 

19.0969 

2.09045 

6.61060 

3.18 

10.1124 

1.78326 

5.63915 

3.78 

14.2884 

1.94422 

6.14817 

4.38 

19.1844 

2.09284 

6.61816 

8.19 

10.1761 

1.78606 

5.64801 

3.79 

14.3641 

1.94679 

6.15630 

4.39 

19.2721 

2.09523 

6.62571 

3.20 

10.2400 

1.78885 

5.65685 

3.80 

14.4400 

1.94936 

6.16441 

4.40 

19.3600 

2.09762 

6.63325 

3.21 

10.3041 

1.79165 

5.66569 

3.81 

14.5161 

1.95192 

6.17252 

4.41 

19.4481 

2.10000 

6.64078 

3.22 

10.3684 

1.79444 

5.67450 

3.82 

14.5924 

1.95448 

6.18061 

4.42 

19.5364 

2.10238 

6.64831 

3.23 

10.4329 

1.79722 

5.68331 

3.83 

14.6689 

1.95704 

6.18870 

4.43 

19.6249 

2.10476 

6.65582 

3.24 

10.4976 

1.80000 

5.69210 

3.84 

14.7456 

1.95959 

6.19677 

4.44 

19.7136 

2.10713 

6.66333 

3.25 

10.5625 

1.80278 

5.70088 

3.85 

14.8225 

1.96214 

6.20484 

4.45 

19.8025 

2.10950 

6.67083 

3.26 

10.6276 

1.80555 

5.70964 

3.86 

14.8996 

1.96469 

6.21289 

4.46 

19.8916 

2.11187 

6.67832 

3.27 

10.6929 

1.80831 

5.71839 

3.87 

14.9769 

1.96723 

*6.22093 

4.47 

19.9809 

2.11424 

6.68581 

3.28 

10.7584 

1.81108 

5.72713 

3.88 

15.0544 

1.96977 

6.22896 

4.48 

20.0704 

2.11660 

6.69328 

3.29 

10.8241 

1.81384 

5.73585 

3.89 

15.1321 

1.97231 

6.23699 

4.49 

20.1601 

2.11896 

6.70075 T 

3.30 

10.8900 

1.81659 

5.74456 

3.90 

15.2100 

1.97484 

6.24500 

4.50 

20.2500 

2.12132 

6.70820 

3.31 

10.9561 

1.81934 

5.75326 

3.91 

15.2881 

1.97737 

6.25300 

4.51 

20.3401 

2.12368 

6.71565 

3.32 

11.0224 

1.82209 

5.76194 

3.92 

15.3664 

1.97990 

6.26099 

4.52 

20.4304 

2.12603 

6.72309 

3.33 

11.0889 

1.82483 

5.77062 

3.93 

15.4449 

1.98242 

6.26897 

4.53 

20.5209 

2.12838 

6.73053 

3.34 

11.1556 

1.82757 

5.77927 

3.94 

15.5236 

1.98494 

6.27694 

4.54 

20.6116 

2.13073 

6.73795 

3.35 

11.2225 

1.83030 

5.78792 

3.95 

15.6025 

1.98746 

6.28490 

4.55 

20.7025 

2.13307 

6.74537 

3.36 

11.2896 

1.83303 

5.79655 

3.96 

15.6816 

1.98997 

6.29285 

4.56 

20.7936 

2.13542 

6.75278 

3.37 

11.3569 

1,.83576 

5.80517 

3.97 

15.7609 

1.99249 

6.30079 

4.57 

20.8849 

2.13776 

6.76018 

3.38 

11.4244 

1.83848 

5.81378 

3.98 

15.8404 

1.99499 

6.30872 

4.58 

20.9764 

2.14009 

6.76757 

3.39 

11.4921 

1.84120 

5.82237 

3.99 

15.9201 

1.99750 

6.31664 

4.59 

21.0681 

2.14243 

6.77495 

3.40 

11.5600 

1.84391 

5.83095 

4.00 

16.0000 

2.00000 

6.32456 

4.60 

21.1600 

2.14476 

6.78233 

N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

V ION 


[H6] 






































































































XII. SQUARES AND SQUARE ROOTS 


N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

V ION 

N 

N 2 

Vn 

VlON 

4.60 

21.1600 

2.14476 

6.78233 

5.20 

27.0400 

2.28035 

7.21110 

5.80 

33.6400 

2.40832 

7.61577 

4.61 

21.2521 

2.14709 

6.78970 

5.21 

27.1441 

2.28254 

7.21803 

5.81 

33.7561 

2.41039 

7.62234 

4.62 

21.3444 

2.14942 

6.79706 

5.22 

27.2484 

2.28473 

7.22496 

5.82 

33.8724 

2.41247 

7.62889 

4.63 

21.4369 

2.15174 

6.80441 

5.23 

27.3529 

2.28692 

7.23187 

5.83 

33.9889 

2.41454 

7.63544 

4.64 

21.5296 

2.15407 

6.81175 

5.24 

27.4576 

2.28910 

7.23878 

5.84 

34.1056 

2.41661 

7.64199 

4.65 

21.6225 

2.15639 

6.81909 

5.25 

27.5625 

2.29129 

7.24569 

5.85 

34.2225 

2.41868 

7.64853 

4.66 

21.7156 

2.15870 

6.82642 

5.26 

27.6676 

2.29347 

7.25259 

5.86 

34.3396 

2.42074 

7.65506 

4.67 

21.8089 

2.16102 

6.83374 

5.27 

27.7729 

2.29565 

7.25948 

5.87 

34.4569 

2.42281 

7.66159 

4.68 

21.9024 

2.16333 

6.84105 

5.28 

27.8784 

2.29783 

7.26636 

5.88 

34.5744 

2.42487 

7.66812 

4.69 

21.9961 

2.16564 

6.84836 

5.29 

27.9841 

2.30000 

7.27324 

5.89 

34.6921 

2.42693 

7.67463 

4.70 

22.0900 

2.16795 

6.85565 

5.30 

28.0900 

2.30217 

7.28011 

5.90 

34.8100 

2.42899 

7.68115 

4.71 

22.1841 

2.17025 

6.86294 

5.31 

28.1961 

2.30434 

7.28697 

5.91 

34.9281 

2.43105 

7.68765 

4.72 

22.2784 

2.17256 

6.87023 

5.32 

28.3024 

2.30651 

7.29383 

5.92 

35.0464 

2.43311 

7.69415 

4.73 

22.3729 

2.17486 

6.87750 

5.33 

28.4089 

2.30868 

7.30068 

5.93 

35.1649 

2.43516 

7.70065 

4.74 

22.4676 

2.17715 

6.88477 

5.34 

28.5156 

2.31084 

7.30753 

5.94 

35.2836 

2.43721 

7.70714 

4.75 

22.5625 

2.17945 

6.89202 

5.35 

28.6225 

2.31301 

7.31437 

5.95 

35.4025 

2.43926 

7.71362 

4.76 

22.6576 

2.18174 

6.89928 

5.36 

28.7296 

2.31517 

7.32120 

5.96 

35.5216 

2.441^1 

7.72010 

4.77 

22.7529 

2.18403 

6.90652 

5.37 

28.8369 

2.31733 

7.32803 

5.97 

35.6409 

2.44336 

7.72658 

4.78 

22.8484 

2.18632 

6.91375 

5.38 

28.9444 

2.31948 

7.33485 

5.98 

35.7604 

2.44540 

7.73305 

4.79 

22.9441 

2.18861 

6.92098 

5.39 

29.0521 

2.32164 

7.34166 

5.99 

35.8801 

2.44745 

7.73951 

4.80 

23.0400 

2.19089 

6.92820 

5.40 

29.1600 

2.32379 

7.34847 

6.00 

36.0000 

2.44949 

7.74597 

4.81 

23.1361 

2.19317 

6.93542 

5.41 

29.2681 

2.32594 

7.35527 

6.01 

36.1201 

2.45153 

7.75242 

4.82 

23.2324 

2.19545 

6.94262 

5.42 

29.3764 

2.32809 

7.36206 

6.02 

36.2404 

2.45357 

7.75887 

4.83 

23.3289 

2.19773 

6.94982 

5.43 

29.4849 

2.33024 

7.36885 

6.03 

36.3609 

2.45561 

7.76531 

4.84 

23.4256 

2.20000 

6.95701 

5.44 

29.5936 

2.33238 

7.37564 

6.04 

36.4816 

2.45764 

7.77174 

4.85 

23.5225 

2.20227 

6.96419 

5.45 

29.7025 

2.33452 

7.38241 

6.05 

36.6025 

2.45967 

7.77817 

4.86 

23.6196 

2.20454 

6.97137 

5.46 

29.8116 

2.33666 

7.38918 

6.06 

36.7236 

2.46171 

7.78460 

4.87 

23.7169 

2.20681 

6.97854 

5.47 

29.9209 

2.33880 

7.39594 

6.07 

36.8449 

2.46374 

7.79102 

4.88 

23.8144 

2.20907 

6.98570 

5.48 

30.0304 

2.34094 

7.40270 

6.08 

36.9664 

2.46577 

7.79744 

4.89 

23.9121 

2.21133 

6.99285 

5.49 

30.1401 

2.34307 

7.40945 

6.09 

37.0881 

2.46779 

7.80385 

4.90 

24.0100 

2.21359 

7.00000 

5.50 

30.2500 

2.34521 

7.41620 

6.10 

37.2100 

2.46982 

7.81025 

4.91 

24.1081 

2.21585 

7.00714 

5.51 

30.3601 

2.34734 

7.42294 

6.11 

37.3321 

2.47184 

7.81665 

4.92 

24.2064 

2.21811 

7.01427 

5.52 

30.4704 

2.34947 

7.42967 

6.12 

37.4544 

2.47386 

7.82304 

4.93 

24.3049 

2.22036 

7.02140 

5.53 

30.5809 

2.35160 

7.43640 

6.13 

37.5769 

2.47588 

7.82943 

4.94 

24.4036 

2.22261 

7.02851 

5.54 

30.6916 

2.35372 

7.44312 

6.14 

37.6996 

2.47790 

7.83582 

4.95 

24.5025 

2.22486 

7.03562 

5.55 

30.8025 

2.35584 

7.44983 

6.15 

37.8225 

2.47992 

7.84219 

4.96 

24.6016 

2.22711 

7.04273 

5.56 

30.9136 

2.35797 

7.45654 

6.16 

37.9456 

2.48193 

7.84857 

4.97 

24.7009 

2.22935 

7.04982 

5.57 

31.0249 

2.36008 

7.46324 

6.17 

38.0689 

2.48395 

7.85493 

4.98 

24.8004 

2.23159 

7.05691 

5.58 

31.1364 

2.36220 

7.46994 

6.18 

38.1924 

2.48596 

7.86130 

4.99 

24.9001 

2.23383 

7.06399 

5.59 

31.2481 

2.36432 

7.47663 

6.19 

38.3161 

2.48797 

7.86766 

5.00 

25.0000 

2.23607 

7.07107 

5.60 

31.3600 

2.36643 

7.48331 

6.20 

38.4400 

2.48998 

7.87401 

5.01 

25.1001 

2.23830 

7.07814 

5.61 

31.4721 

2.36854 

7.48999 

6.21 

38.5641 

2.49199 

7.88036 

5.02 

25.2004 

2.24054 

7.08520 

5.62 

31.5844 

2.37065 

7.49667 

6.22 

38.6884 

2.49399 

7.88670 

5.03 

25.3009 

2.24277 

7.09225 

5.63 

31.6969 

2.37276 

7.50333 

6.23 

38.8129 

2.49600 

7.89303 

5.04 

25.4016 

2.24499 

7.09930 

5.64 

31.8096 

2.37487 

7.50999 

6.24 

38.9376 

2.49800 

7.89937 

5.05 

25.5025 

2.24722 

7.10634 

5.65 

31.9225 

2.37697 

7.51665 

6.25 

39.0625 

2.50000 

7.90569 

5.06 

25.6036 

2.24944 

7.11337 

5.66 

32.0356 

2.37908 

7.52330 

6.26 

39.1876 

2.50200 

7.91202 

5.07 

25.7049 

2.25167 

7.12039 

5.67 

32.1489 

2.38118 

7.52994 

6.27 

39.3129 

2.50400 

7.91833 

5.08 

25.8064 

2.25389 

7.12741 

5.68 

32.2624 

2.38328 

7.53658 

6.28 

39.4384 

2.50599 

7.92465 

5.09 

25.9081 

2.25610 

7.13442 

5.69 

32.3761 

2.38537 

7.54321 

6.29 

39.5641 

2.50799 

7.93095 

5.10 

26.0100 

2.25832 

7.14143 

5.70 

32.4900 

2.38747 

7.54983 

6.30 

39.6900 

2.50998 

7.93725 

5.11 

26.1121 

2.26053 

7.14843 

5.71 

32.6041 

2.38956 

7.55645 

6.31 

39.8161 

2.51197 

7.94355 

5.12 

26.2144 

2.26274 

7.15542 

5.72 

32.7184 

2.39165 

7.56307 

6.32 

39.9424 

2.51396 

7.94984 

5.13 

26.3169 

2.26495 

7.16240 

5.73 

32.8329 

2.39374 

7.56968 

6.33 

40.0689 

2.51595 

7.95613 

5.14 

26.4196 

2.26716 

7.16938 

5.74 

32.9476 

2.39583 

7.57628 

6.34 

40.1956 

2.51794 

7.96241 

5.15 

26.5225 

2.26936 

7.17635 

5.75 

33.0625 

2.39792 

7.58288 

6.35 

40.3225 

2.51992 

7.96869 

5.16 

26.6256 

2.27156 

7.18331 

5.76 

33.1776 

2.40000 

7.58947 

6.36 

40.4496 

2.52190 

7.97496 

5.17 

26.7289 

2.27376 

7.19027 

5.77 

33.2929 

2.40208 

7.59605 

6.37 

40.5769 

2.52389 

7.98123 

5.18 

26.8324 

2.27596 

7.19722 

5.78 

33.4084 

2.40416 

7.60263 

6.38 

40.7044 

2.52587 

7.98749 

5.19 

26.9361 

2.27816 

7.20417 

5.79 

33.5241 

2.40624 

7.60920 

6.39 

40.8321 

2.52784 

7.99375 

5.20 

27.0400 

2.28035 

7.21110 

5.80 

33.6400 

2.40832 

7.61577 

6.40 

40.9600 

2.52982 

8.00000 

N 

N 2 

Vn 

VlO N 

N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

VlON 


[ 117 ] 





























































































XII. SQUARES AND SQUARE ROOTS 


N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

Vl ON 

N 

N 2 

Vn 

V ION 

6.40 

40.9600 

2.52982 

8.00000 

7.00 

49.0000' 

2.64575 

8.36660 

7.60 

57.7600 

2.75681 

8.71780 

6.41 

41.0881 

2.53180 

8.00625 

7.01 

49.1401 

2.64764 

8.37257 

7.61 

57.9121 

2.75862 

8.72353 

6.42 

41.2164 

2.53377 

8.01249 

7.02 

49.2804 

2.64953 

8.37854 

7.62 

58.0644 

2.76043 

8.72926 

6.43 

41.3449 

2.53574 

8.01873 

7.03 

49.4209 

2.65141 

8.38451 

7.63 

58.2169 

2.76225 

8.73499 

6.44 

41.4736 

2.53772 

8.02496 

7.04 

49.5616 

2.65330 

8.39047 

7.64 

58.3696 

2.76405 

8.74071 

6.45 

41.6025 

2.53969 

8.03119 

7.05 

49.7025 

2.65518 

8.39643 

7.65 

58.5225 

2.76586 

8.74643 

6.46 

41.7316 

2.54165 

8.03741 

7.06 

49.8436 

2.65707 

8.40238 

7.66 

58.6756 

2.76767 

8.75214 

6.47 

41.8609 

2.54362 

8.04363 

7.07 

49.9849 

2.65895 

8.40833 

7.67 

58.8289 

2.76948 

8.75785 

6.48 

41.9904 

2.54558 

8.04984 

7.08 

50.1264 

2.66083 

8.41427 

7.68 

58.9824 

2.77128 

8.76356 

6.49 

42.1201 

2.54755 

8.05605 

7.09 

50.2681 

2.66271 

8.42021 

7.69 

59.1361 

2.77308 

8.76926 

6.50 

42.2500 

2.54951 

8.06226 

7.10 

50.4100 

2.66458 

8.42615 

7.70 

59.2900 

2.77489 

8.77496 

6.51 

42.3801 

2.55147 

8.06846 

7.11 

50.5521 

2.66646 

8.43208 

7.71 

59.4441 

2.77669 

8.78066 

6.52 

42.5104 

2.55343 

8.07465 

7.12 

50.6944 

2.66833 

8.43801 

7.72 

59.5984 

2.77849 

8.78635 

6.53 

42.6409 

2.55539 

8.08084 

7.13 

50.8369 

2.67021 

8.44393 

7.73 

59.7529 

2.78029 

8.79204 

6.54 

42.7716 

2.55734 

8.0S703 

7.14 

50.9796 

2.67208 

8.44985 

7.74 

59.9076 

2.78209 

8.79773 

6.55 

42.9025 

2.55930 

8.09321 

7.15 

51.1225 

2.67395 

8.45577 

7.75 

60.0625 

2.78388 

8.80341 

6.56 

43.0336 

2.56125 

8.09938 

7.16 

51.2656 

2.67582 

8.46168 

7.76 

60.2176 

2.78568 

8.80909 

6.57 

43.1649 

2.56320 

8.10555 

7.17 

51.4089 

2.67769 

8.46759 

7.77 

60.3729 

2.78747 

8.81476 

6.58 

43.2964 

2.56515 

8.11172 

7.18 

51.5524 

2.67955 

8.47349 

7.78 

60.5284 

2.78927 

8.82043 

6.59 

43.4281 

2.56710 

8.11788 

7.19 

51.6961 

2.68142 

8.47939 

7.79 

60.6841 

2.79106 

8.82610 

6.60 

43.5600 

2.56905 

8.12404 

7.20 

51.8400 

2.68328 

8.48528 

7.80 

60.8400 

2.79285 

8.83176 

6.61 

43.6921 

2.57099 

8.13019 

7.21 

51.9841 

2.68514 

8.49117 

7.81 

60.9961 

2.79464 

8.83742 

6.62 

43.8244 

2.57294 

8.13634 

7.22 

52.1284 

2.68701 

8.49706 

7.82 

61.1524 

2.79643 

8.84308 

6.63 

43.9569 

2.57488 

8.14248 

7.23 

52.2729 

2.68887 

8.50294 

7.83 

61.3089 

2.79821 

8.84873 

6.64 

44.0896 

2.57682 

8.14862 

7.24 

52.4176 

2.69072 

8.50882 

7.84j 

61.4656 

2.80000 

8.85438 

6.65 

44.2225 

2.57876 

8.15475 

7.25 

52.5625 

2.69258 

8.51469 

7.85 

61.6225 

2.80179 

8.86002 

6.66 

44.3556 

2.58070 

8.16088 

7.26 

52.7076 

2.69444 

8.52056 

7.86 

61.7796 

2.80357 

8.86566 

6.67 

44.4889 

2.58263 

8.16701 

7.27 

52.8529 

2.69629 

8.52643 

7.87 

61.9369 

2.80535 

8.87130 

6.68 

44.6224 

2.58457 

8.17313 

7.28 

52.9984 

2.69815 

8.53229 

7.88 

62.0944 

2.80713 

8.87694 

6.69 

44.7561 

2.58650 

8.17924 

7.29 

53.1441 

2.70000 

8.53815 

7.89 

62.2521 

2.80891 

8.88257 

6.70 

44.8900 

2.58844 

8.18535 

7.30 

53.2900 

2.70185 

8.54400 

7.90 

62.4100 

2.81069 

8.88819 

6.71 

45.0241 

2.59037 

8.19146 

7.31 

53.4361 

2.70370 

8.54985 

7.91 

62.5681 

2.81247 

8.89382 

6.72 

45.1584 

2.59230 

8.19756 

7.32 

53.5824 

2.70555 

8.55570 

7.92 

62.7264 

2.81425 

8.89944 

6.73 

45.2929 

2.59422 

8.20366 

7.33 

53.7289 

2.70740 

8.56154 

7.93 

62.8849 

2.81603 

8.90505 

6.74 

45.4276 

2.59615 

8.20975 

7.34 

53.8756 

2.70924 

8.56738 

7.94 

63.0436 

2.81780 

8.91067 

6.75 

45.5625 

2.59808 

8.21584 

7.35 

54.0225 

2 71109 

8.57321 

7.95 

63.2025 

2.81957 

8.91628 

6.76 

45.6976 

2.60000 

8.22192 

7.36 

54.1696 

2.71293 

8.57904 

7.96 

63.3616 

2.82135 

8.92188 

6.77 

45.8329 

2.60192 

8.22800 

7.37 

54.3169 

2.71477 

8.58487 

7.97 

63.5209 

2.82312 

8.92749 

6.78 

45.9684 

2.60384 

8.23408 

7.38 

54.4644 

2.71662 

8.59069 

7.98 

63.6804 

2.82489 

8.93308 

6.79 

46.1041 

2.60576 

8.24015 

7.39 

54.6121 

2.71846 

8.59651 

7.99 

63.8401 

2.82666 

8.93868 

6.80 

46.2400 

2.60768 

8.24621 

7.40 

54.7600 

2.72029 

8.60233 

8.00 

64.0000 

2.82843 

8.94427 

6.81 

46.3761 

2.60960 

8.25227 

7.41 

54.9081 

2.72213 

8.60814 

8.01 

64.1601 

2.83019 

8.94986 

6.82 

46.5124 

2.61151 

8.25833 

7.42 

55.0564 

2.72397 

8.61394 

8.02 

64.3204 

2.83196 

8.95545 

6.83 

46.6489 

2.61343 

8.26438 

7.43 

55.2049 

2.72580 

8.61974 

8.03 

64.4809 

2.83373 

8.96103 

6.84 

46.7856 

2.61534 

8.27043 

7.44 

55.3536 

2.72764 

8.62554 

8.04 

64.6416 

2.83549 

8.96660 

6.85 

46.9225 

2.61725 

8.27647 

7.45 

55.5025 

2.72947 

8.63134 

8.05 

64.8025 

2.83725 

8.97218 

6.86 

47.0596 

2.61916 

8.28251 

7.46 

55.6516 

2.73130 

8.63713 

8.06 

64.9636 

2.83901 

8.97775 

6.87 

47.1969 

2.62107 

8.28855 

7.47 

55.8009 

2.73313 

8.64292 

8.07 

65.1249 

2.84077 

8.98332 

6.88 

47.3344 

2.62298 

8.29458 

7.48 

55.9504 

2.73496 

8.64870 

8.08 

65.2864 

2.84253 

8.98888 

6.89 

47.4721 

2.62488 

8.30060 

7.49 

56.1001 

2.73679 

8.65448 

8.09 

65.4481 

2.84429 

8.99444 

6.90 

47.6100 

2.62679 

8.30662 

7.50 

56.2500 

2.73861 

8.66025 

8.10 

65.6100 

2.84605 

9.00000 

6.91 

47.7481 

2.62869 

8.31264 

7.51 

56.4001 

2.74044 

8.66603 

8.11 

65.7721 

2.84781 

9.00555 

6.92 

47.8864 

2.63059 

8.31865 

7.52 

56.5504 

2.74226 

8.67179 

8.12 

65.9344 

2.84956 

9.01110 

6.93 

48.0249 

2.63249 

8.32466 

7.53 

56.7009 

2.74408 

8.67756 

8.13 

66.0969 

2.85132 

9.01665 

6.94 

48.1636 

2.63439 

8.33067 

7.54 

56.8516 

2.74591 

8.68332 

8.14 

66.2596 

2.85307 

9.02219 

6.95 

48.3025 

2.63629 

8.33667 

7.55 

57.0025 

2.74773 

8.68907 

8.15 

66.4225 

2.85482 

9.02774 

6.96 

48.4416 

2.63818 

8.34266 

7.56 

57.1536 

2.74955 

8.69483 

8.16 

66.5856 

2.85657 

9.03327 

6.97 

48.5809 

2.64008 

8.34865 

7.57 

57.3049 

2.75136 

8.70057 

8.17 

66.7489 

2.85832 

9.03881 

6.98 

48.7204 

2.64197 

8.35464 

7.58 

57.4564 

2.75318 

8.70632 

8.18 

66.9124 

2.86007 

9.04434 

6.99 

48.8601 

2.64386 

8.36062 

7.59 

57.6081 

2.75500 

8.71206 

8.19 

67.0761 

2.86182 

9.04986 

7.00 

49.0000 

2.64575 

8.36660 

7.60 

57.7600 

2.75681 

8.71780 

8.20 

67.2400 

2.86356 

9.05539 

N 

N 2 

Vn 

V ION 

N 

N* 

Vn 

V ION 

N 

N 2 

Vn 

V ION 


CHS] 
































































































XII. SQUARES AND SQUARE ROOTS 


N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

VlON 

N 

N 2 

Vn 

VlON 

8.20 

67.2400 

2.86356 

9.05539 

8.80 

77.4400 

2.96648 

9.38083 

9.40 

88.3600 

3.06594 

9.69536 

8.21 

67.4041 

2.86531 

9.06091 

8.81 

77.6161 

2.96816 

9.38616 

9.41 

88.5481 

3.06757 

9.70052 

8.22 

67.5684 

2.86705 

9.06642 

8.82 

77.7924 

2.96985 

9.39149 

9.42 

88.7364 

3.06920 

9.70567 

8.23 

67.7329 

2.86880 

9.07193 

8.83 

77.9689 

2.97153 

9.39681 

9.43 

88.9249 

3.07083 

9.71082 

8.24 

67.8976 

2.87054 

9.07744 

8.84 

78.1456 

2.97321 

9.40213 

9.44 

89.1136 

3.07246 

9.71597 

8.25 

68.0625 

2.87228 

9.08295 

8.85 

78.3225 

2.97489 

9.40744 

9.45 

89.3025 

3.07409 

9.72111 

8.26 

68.2276 

2.87402 

9.08845 

8.86 

78.4996 

2.97658 

9.41276 

9.46 

89.4916 

3.07571 

9.72625 

8.27 

68.3929 

2.87576 

9.09395 

8.87 

78.6769 

2.97825 

9.41807 

9.47 

89.6809 

3.07734 

9.73139 

8.28 

68.5584 

2.87750 

9.09945 

8.88 

78.8544 

2.97993 

9.42338 

9.48 

89.8704 

3.07896 

9.73653 

8.29 

68.7241 

2.87924 

9.10494 

8.89 

79.0321 

2.98161 

9.42868 

9.49 

90.0601 

3.08058 

9.74166 

8.30 

68.8900 

2.88097 

9.11043 

8.90 

79.2100 

2.98329 

9.43398 

9.50 

90.2500 

3.08221 

9.74679 

8.31 

69.0561 

2.88271 

9.11592 

8.91 

79.3881 

2.98496 

9.43928 

9.51 

90.4401 

3.08383 

9.75192 

8.32 

69.2224 

2.88444 

9.12140 

8.92 

79.5664 

2.98664 

9.44458 

9.52 

90.6304 

3.08545 

9.75705 

8.33 

69.3889 

2.88617 

9.12688 

8.93 

79.7449 

2.98831 

9.44987 

9.53 

90.8209 

3.08707 

9.76217 

8.34 

69.5556 

2.88791 

9.13236 

8.94 

79.9236 

2.98998 

9.45516 

9.54 

91.0116 

3.08869 

9.76729 

8.35 

69.7225 

2.88964 

9.13783 

8.95 

80.1025 

2.99166 

9.46044 

9.55 

91.2025 

3.09031 

9.77241 

8.36 

69.8896 

2.89137 

9.14330 

8.96 

80.2816 

2.99333 

9.46573 

9.56 

91.3936 

3.09192 

9.77753 

8.37 

70.0569 

2.89310 

9.14877 

8.97 

80.4609 

2.99500 

9.47101 

9.57 

91.5849 

3.09354 

9.78264 

8.38 

70.2244 

2.89482 

9.15423 

8.98 

80.6404 

2.99666 

9.47629 

9.58 

91.7764 

3.09516 

9.78775 

8.39 

70.3921 

2.89655 

9.15969 

8.99 

80.8201 

2.99833 

9.48156 

9.59 

91.9681 

3.09677 

9.79285 

8.40 

70.5600 

2.89828 

9.16515 

9.00 

81.0000 

3.00000 

9.48683 

9.60 

92.1600 

3.09839 

9.79796 

8.41 

70.7281 

2.90000 

9.17061 

9.01 

81.1801 

3.00167 

9.49210 

9.61 

92.3521 

3.10000 

9.80306 

8.42 

70.8964 

2.90172 

9.17606 

9.02 

81.3604 

3.00333 

9.49737 

9.62 

92.5444 

3.10161 

9.80816 

8.43 

71.0649 

2.90345 

9.18150 

9.03 

81.5409 

3.00500 

9.50263 

9.63 

92.7369 

3.10322 

9.81326 

8.44 

71.2336 

2.90517 

9.18695 

9.04 

81.7216 

3.00666 

9.50789 

9.64 

92.9296 

3.10483 

9.81835 

8.45 

71.4025 

2.90689 

9.19239 

9.05 

81.9025 

3.00832 

9.51315 

9.65 

93.1225 

3.10644 

9.82344 

8.46 

71.5716 

2.90861 

9.19783 

9.06 

82.0836 

3.00998 

9.51840 

9.66 

93.3156 

3.10805 

9.82853 

8.47 

71.7409 

2.91033 

9.20326 

9.07 

82.2649 

3.01164 

9.52365 

9.67 

93.5089 

3.10966 

9.83362 

8.48 

71.9104 

2.91204 

9.20869 

9.08 

82.4464 

3.01330 

9.52890 

9.68 

93.7024 

3.11127 

9.83870 

8.49 

72.0801 

2.91376 

9.21412 

9.09 

82.6281 

3.01496 

9.53415 

9.69 

93.8961 

3.11288 

9.84378 

8.50 

72.2500 

2.91548 

9.21954 

9.10 

82.8100 

3.01662 

9.53939 

9.70 

94.0900 

3.11448 

9.84886 

8.51 

72.4201 

2.91719 

9.22497 

9.11 

82.9921 

3.01828 

9.54463 

9.71 

94.2841 

3.11609 

9.85393 

8.52 

72.5904 

2.91890 

9.23038 

9.12 

83.1744 

3.01993 

9.54987 

9.72 

94.4784 

3.11769 

9.85901 

8.53 

72.7609 

2.92062 

9.23580 

9.13 

83.3569 

3.02159 

9.55510 

9.73 

94.6729 

3.11929 

9.86408 

8.54 

72.9316 

2.92233 

9.24121 

9.14 

83.5396 

3.02324 

9.56033 

9.74 

94.8676 

3.12090 

9.86914 

8.55 

73.1025 

2.92404 

9.24662 

9.15 

83.7225 

3.02490 

9.56556 

9.75 

95.0625 

3.12250 

9.87421 

8.56 

73.2736 

2.92575 

9.25203 

9.16 

83.9056 

3.02655 

9.57079 

9.76 

95.2576 

3.12410 

9.87927 

8.57 

73.4449 

2.92746 

9.25743 

9.17 

84.0889 

3.02820 

9.57601 

9.77 

95.4529 

3.12570 

9.88433 

8.58 

73.6164 

2.92916 

9.26283 

9.18 

84.2724 

3.02985 

9.58123 

9.78 

95.6484 

3.12730 

9.88939 

8.59 

73.7881 

2.93087 

9.26823 

9.19 

84.4561 

3.03150 

9.58645 

9.79 

95.8441 

3.12890 

9.89444 

8.60 

73.9600 

2.93258 

9.27362 

9.20 

84.6400 

3.03315 

9.59166 

9.80 

96.0400 

3.13050 

9.89949 

8.61 

74.1321 

2.93428 

9.27901 

9.21 

84.8241 

3.03480 

9.59687 

9.81 

96.2361 

3.13209 

9.90454 

8.62 

74.3044 

2.93598 

9.28440 

9.22 

85.0084 

3.03645 

9.60208 

9.82 

96.4324 

3.13369 

9.90959 

8.63 

74.4769 

2.93769 

9.28978 

9.23 

85.1929 

3.03809 

9.60729 

9.83 

96.6289 

3.13528 

9.91464 

8.64 

74.6496 

2.93939 

9.29516 

9.24 

85.3776 

3.03974 

9.61249 

9.84 

96.8256 

3.13688 

9.91968 

8.65 

74.8225 

2.94109 

9.30054 

9.25 

85.5625 

3.04138 

9.61769 

9.85 

97.0225 

3.13847 

9.92472 

8.66 

74.9956 

2.94279 

9.30591 

9.26 

85.7476 

3.04302 

9.62289 

9.86 

97.2196 

3.14006 

9.92975 

8.67 

75.1689 

2.94449 

9.31128 

9.27 

85.9329 

3.04467 

9.62808 

9.87 

97.4169 

3.14166 

9.93479 

8.68 

75.3424 

2.94618 

9.31665 

9.28 

86 1184 

3.04631 

9.63328 

9.88 

97.6144 

3.14325 

9.93982 

8.69 

75.5161 

2.94788 

9.32202 

9.29 

86.3041 

3.04795 

9.63846 

9.89 

97.8121 

3.14484 

9.94485 

8.70 

75.6900 

2.94958 

9.32738 

9.30 

86.4900 

3.04959 

9.64365 

9.90 

98.0100 

3.14643 

9.94987 

8.71 

75.8641 

2.95127 

9.33274 

9.31 

86.6761 

3.05123 

9.64883 

9.91 

98.2081 

3.14802 

9.95490 

8.72 

76.03S4 

2.95296 

9.33809 

9.32 

86.8624 

3.05287 

9.65401 

9.92 

98.4064 

3.14960 

9.95992 

8.73 

76.2129 

2.95466 

9.34345 

9.33 

87.0489 

3.05450 

9.65919 

9.93 

98.6049 

3.15119 

9.96494 

8.74 

76.3876 

2.95635 

9.34880 

9.34 

87.2356 

3.05614 

9.66437 

9.94 

98.8036 

3.15278 

9.96995 

8.75 

76.5625 

2.95804 

9.35414 

9.35 

87.4225 

3.05778 

9.66954 

9.95 

99.0025 

3.15436 

9.97497 

8.76 

76.7376 

2.95973 

9.35949 

9.36 

87.6096 

3.05941 

9.67471 

9.96 

99.2016 

3.15595 

9.97998 

8.77 

76.9129 

2.96142 

9.36483 

9.37 

87.7969 

3.06105 

9.67988 

9.97 

99.4009 

3.15753 

9.98499 

8.78 

77.0884 

2.96311 

9.37017 

9.38 

87.9844 

3.06268 

9.68504 

9.98 

99.6004 

3.15911 

9.98999 

8.79 

77.2641 

2.96479 

9.37550 

9.39 

88.1721 

3.06431 

9.69020 

9.99 

99.8001 

3.16070 

9.99500 

8.80 

77.4400 

2.96648 

9.38083 

9.40 

88.3600 

3.06594 

9.69536 

10.00 

100.000 

3.16228 

10.0000 

N 

N 2 

Vn 

V ION 

N 

N 2 

Vn 

V ION 

N 

N 2 

Vn 

VlON 


[ 119 ] 























































































XIII. RADIAN MEASURE: VALUES OF FUNCTIONS 


a 

Rad. 

Degrees 
in a 

Sin a 

Cos a 

Tan a 


a 

Rad. 

Degrees 
in a 

Sin a 

Cos a 

Tan a 

.00 

0° 00.0' 

.00000 

1.0000 

.00000 


.60 

34° 22.6' 

.56464 

.82534 

.68414 

.01 

0° 34.4' 

.01000 

.99995 

.01000 


.61 

34° 57.0' 

.57287 

.81965 

.69892 

.02 

1° 08.8' 

.02000 

.99980 

.02000 


.62 

35° 31.4' 

.58104 

.81388 

.71391 

.03 

1° 43.1' 

.03000 

.99955 

.03001 


.63 

36° 05.8' 

.58914 

.80803 

.72911 

.04 

2° 17.5' 

.03999 

.99920 

.04002 


.64 

36° 40.2' 

.59720 

.80210 

.74454 

.05 

2° 51.9' 

.04998 

.99875 

.05004 


.65 

37° 14.5' 

.60519 

.79608 

.76020 

.06 

3° 26.3' 

.05996 

.99820 

.06007 


.66 

37° 48.9' 

.61312 

.78999 

.77610 

.07 

4° 00.6' 

.06994 

.99755 

.07011 


.67 

38° 23.3' 

.62099 

.78382 

.79225 

.08 

4° 35.0' 

.07991 

.99680 

.08017 


.68 

38° 57.7' 

.62879 

.77757 

.80866 

.09 

5° 09.4' 

.08988 

.99595 

.09024 


.69 

39° 32.0' 

.63654 

.77125 

.82534 

.10 

5° 43.8' 

.09983 

.99500 

.10033 


.70 

40° 06.4' 

.64422 

.76484 

.84229 

.11 

6° 18.2' 

.10978 

.99396 

.11045 


.71 

40° 40.8' 

.65183 

.75836 

.85953 

.12 

6° 52.5' 

.11971 

.99281 

.12058 


.72 

41° 15.2' 

.65938 

.75181 

.87707 

.13 

7° 26.9' 

.12963 

.99156 

.13074 


.73 

41° 49.6' 

.66687 

.74517 

.89492 

.14 

8° 01.3' 

.13954 

.99022 

.14092 


.74 

42° 23.9' 

.67429 

.73847 

.91309 

.15 

8° 35.7' 

.14944 

.98877 

.15114 


.75 

42° 58.3' 

.68164 

.73169 

.93160 

.16 

9° 10.0' 

.15932 

.98723 

.16138 


.76 

43° 32.7' 

.68892 

.72484 

.95045 

.17 

9° 44.4' 

.16918 

.98558 

.17166 


.77 

44° 07.1' 

.69614 

.71791 

.96967 

.18 

10° 18.8' 

.17903 

.98384 

.18197 


.78 

44° 41.4' 

.70328 

.71091 

.98926 

.19 

10° 53.2' 

.18886 

.98200 

.19232 


.79 

45° 15.8' 

.71035 

.70385 

1.0092 

.20 

11° 27.5' 

.19867 

.98007 

.20271 


.80 

45° 50.2' 

.71736 

.69671 

1.0296 

.21 

12° 01.9' 

.20846 

.97803 

.21314 


.81 

46° 24.6' 

.72429 

.68950 

1.0505 

.22 

12° 36.3' 

.21823 

.97590 

.22362 


.82 

46° 59.0' 

.73115 

.68222 

1.0717 

.23 

13° 10.7' 

.22798 

.97367 

.23414 


.83 

47° 33.3' 

.73793 

.67488 

1.0934 

.24 

13° 45.1' 

.23770 

.97134 

.24472 


.84 

48° 07.7' 

.74464 

.66746 

1.1156 

.25 

14° 19.4' 

.24740 

.96891 

.25534 


.85 

48° 42.1' 

.75128 

.65998 

1.1383 

.26 

14° 53.8' 

.25708 

.96639 

.26602 


.86 

49° 16.5' 

.75784 

.65244 

1.1616 

.27 

15° 28.2' 

.26673 

.96377 

.27676 


.87 

49° 50.8' 

.76433 

.64483 

1.1853 

.28 

16° 02.6' 

.27636 

.96106 

.28755 


.88 

50° 25.2' 

.77074 

.63715 

1.2097 

.29 

16° 36.9' 

.28595 

.95824 

.29841 


.89 

50° 59.6' 

.77707 

.62941 

1.2346 

.30 

17° 11.3' 

.29552 

.95534 

.30934 


.90 

51° 34.0' 

.78333 

.62161 

1.2602 

.31 

17° 45.7' 

.30506 

.95233 

.32033 


.91 

52° 08.3' 

.78950 

.61375 

1.2864 

.32 

18° 20.1' 

.31457 

.94924 

.33139 


.92 

52° 42.7' 

.79560 

.60582 

1.3133 

.33 

18° 54.5' 

.32404 

.94604 

.34252 


.93 

53° 17.1' 

.80162 

.59783 

1.3409 

.34 

19° 28.8' 

.33349 

.94275 

.35374 


.94 

53° 51.5' 

.80756 

.58979 

1.3692 

.35 

20° 03.2' 

.34290 

.93937 

.36503 


.95 

54° 25.9' 

.81342 

.58168 

1.3984 

.36 

20° 37.6' 

.35227 

.93590 

.37640 


.96 

55° 00.2' 

.81919 

.57352 

1.4284 

.37 

21° 12.0' 

.36162 

.93233 

.38786 


.97 

55° 34.6' 

.82489 

.56530 

1.4592 

.3.8 

21° 46.3' 

.37092 

.92866 

.39941 


.98 

56° 09.0' 

.83050 

.55702 

1.4910 

.39 

22° 20.7' 

.38019 

.92491 

.41105 


.99 

56° 43.4' 

.83603 

.54869 

1.5237 

.40 

22° 55.1' 

.38942 

.92106 

.42279 


1.00 

57° 17.7' 

.84147 

.54030 

1.5574 

.41 

23° 29.5' 

.39861 

.91712 

.43463 


1.01 

57° 52.1' 

.84683 

.53186 

1.5922 

.42 

24° 03.9' 

.40776 

.91309 

.44657 


1.02 

58° 26.5' 

.85211 

.52337 

1.6281 

.43 

24° 38.2' 

.41687 

.90897 

.45862 


1.03 

59° 00.9' 

.85730 

.51482 

1.6652 

.44 

25° 12.6' 

.42594 

.90475 

.47078 


1.04 

59° 35.3' 

.86240 

.50622 

1.7036 

.45 

25° 47.0' 

.43497 

.90045 

.48306 


1.05 

60° 09.6' 

.86742 

.49757 

1.7433 

.46 

26° 21.4' 

.44395 

.89605 

.49545 


1.06 

60° 44.0' 

.87236 

.48887 

1.7844 

.47 

26° 55.7' 

.45289 

.89157 

.50797 


1.07 

61° 18.4' 

.87720 

.48012 

1.8270 

.48 

27° 30.1' 

.46178 

.88699 

.52061 


1.08 

61° 52.8' 

.88196 

.47133 

1.8712 

.49 

28° 04.5' 

.47063 

.88233 

.53339 


1.09 

62° 27.1' 

.88663 

.46249 

1.9171 

.50 

28° 38.9' 

.47943 

.87758 

.54630 


1.10 

63° 01.5' 

.89121 

.45360 

1.9648 

.51 

29° 13.3' 

.48818 

.87274 

.55936 


1.11 

63° 35.9' 

.89570 

.44466 

2.0143 

.52 

29° 47.6' 

.49688 

.86782 

.57256 


1.12 

64° 10.3' 

.90010 

.43568 

2.0660 

.53 

30° 22.0' 

.50553 

.86281 

.58592 


1.13 

64° 44.7' 

.90441 

.42666 

2.1198 

.54 

30° 56.4' 

.51414 

.85771 

.59943 


1.14 

65° 19.0' 

.90863 

.41759 

2.1759 

.55 

31° 30.8' 

.52269 

.85252 

.61311 


1.15 

65° 53.4' 

.91276 

.40849 

2.2345 

.56 

32° 05.1' 

.53119 

.84726 

.62695 


1.16 

66° 27.8' 

.91680 

.39934 

2.2958 

.57 

32° 39.5' 

.53963 

.84190 

.64097 


1.17 

67° 02.2' 

.92075 

.39015 

2.3600 

.58 

33° 13.9' 

.54802 

.83646 

.65517 


1.18 

67° 36.5' 

.92461 

.38092 

2.4273 

.59 

33° 48.3' 

.55636 

.83094 

.66956 


1.19 

68° 10.9' 

.92837 

.37166 

2.4979 

.60 

34° 22.6' 

.56464 

.82534 

.68414 


1.20 

68° 45.3' 

.93204 

.36236 

2.5722 


[ 120 ] 
































































XIII. RADIAN MEASURE 


VALUES OF FUNCTIONS 


a 

Rad. 

Degrees 
in a . 

Sin a 

Cos a 

Tan a 

1.20 

68 ° 45 . 3 ' 

.93204 

.36236 

2.5722 

1.21 

69 ° 19 . 7 ' 

.93562 

.35302 

2.6503 

1.22 

69 ° 54 . 1 ' 

.93910 

.34365 

2.7328 

1.23 

70 ° 28 . 4 ' 

.94249 

.33424 

2.8198 

1.24 

71 ° 02 . 8 ' 

.94578 

.32480 

2.9119 

1.25 

71 ° 37 . 2 ' 

.94898 

.31532 

3.0096 

1.26 

72 ° 11 . 6 ' 

.95209 

.30582 

3.1133 

1.27 

72 ° 45 . 9 ' 

.95510 

.29628 

3.2236 

1.28 

73 ° 20 . 3 ' 

.95802 

.28672 

3.3413 

1.29 

73 ° 54 . 7 ' 

.96084 

.27712 

3.4672 

1.30 

74 ° 29 . 1 ' 

.96356 

.26750 

3.6021 

1.31 

75 ° 03 . 4 ' 

.96618 

.25785 

3.7471 

1.32 

75 ° 37 . 8 ' 

.96872 

.24818 

3.9033 

1.33 

76 ° 12 . 2 ' 

.97115 

.23848 

4.0723 

1.34 

76 ° 46 . 6 ' 

.97348 

.22875 

4.2556 

1.35 

77 ° 21 . 0 ' 

.97572 

.21901 

4.4552 

1.36 

77 ° 55 . 3 ' 

.97786 

.20924 

4.6734 

1.37 

78 ° 29 . 7 ' 

.97991 

.19945 

4.9131 

1.38 

79 ° 04 . 1 ' 

.98185 

.18964 

5.1774 

1.39 

79 ° 38 . 5 ' 

.98370 

.17981 

5.4707 

1.40 

80 ° 12 . 8 ' 

.98545 

.16997 

5.7979 


a 

Rad. 

Degrees 
in a 

Sin a 

Cos a 

Tan a 

1.40 

80 ° 12 . 8 ' 

.98545 

.16997 

5.7979 

1.41 

80 ° 47 . 2 ' 

.98710 

.16010 

6.1654 

1.42 

81 ° 21 . 6 ' 

.98865 

.15023 

6.5811 

1.43 

81 ° 56 . 0 ' 

.99010 

.14033 

7.0555 

1.44 

82 ° 30 . 4 ' 

.99146 

.13042 

7.6018 

1.45 

83 ° 04 . 7 ' 

.99271 

.12050 

8.2381 

1.46 

83 ° 39 . 1 ' 

.99387 

.11057 

8.9886 

1.47 

84 ° 13 . 5 ' 

.99492 

.10063 

9.8874 

1.48 

84 ° 47 . 9 ' 

.99588 

.09067 

10.983 

1.49 

85 ° 22 . 2 ' 

.99674 

.08071 

12.350 

1.50 

85 ° 56 . 6 ' 

.99749 

.07074 

14.101 

1.51 

86 ° 31 . 0 ' 

.99815 

.06076 

16.428 

1.52 

87 ° 05 . 4 ' 

.99871 

.05077 

19.670 

1.53 

87 ° 39 . 8 ' 

.99917 

.04079 

24.498 

1.54 

88 ° 14 . 1 ' 

.99953 

.03079 

32.461 

1.55 

88 ° 48 . 5 ' 

.99978 

.02079 

48.078 

1.56 

89 ° 22 . 9 ' 

.99994 

.01080 

92.620 

1.57 

89 ° 57 . 3 ' 

1.0000 

.00080 

1255.8 

1.58 

90 ° 31 . 6 ' 

.99996 

- .00920 

- 108.65 

1.59 

91 ° 06 . 0 ' 

.99982 

-.01920 

- 52.067 

1.60 

91 ° 40 . 4 ' 

.99957 

- .02920 

- 34.233 


DEGREES IN RADIANS 


1 ° 

0.01745 

16 ° 

0.27925 

31 ° 

0.54105 

46 ° 

0.80285 

61 ° 

1.06465 

76 ° 

1.32645 

2 

0.03491 

17 

0.29671 

32 

0.55851 

47 

0.82030 

62 

1.08210 

77 

1.34390 

3 

0.05236 

18 

0.31416 

33 

0.57596 

48 

0.83776 

63 

1.09956 

78 

1.36136 

4 

0.06981 

19 

0.33161 

34 

0.59341 

49 

0.85521 

64 

1.11701 

79 

1.37881 

5 

0.08727 

20 

0.34907 

35 

0.61087 

50 

0.87266 

65 

1.13446 

80 

1.39626 

6 

0.10472 

21 

0.36652 

36 

0.62832 

51 

0.89012 

66 

1.15192 

81 

1.41372 

7 

0.12217 

22 

0.38397 

37 

0.64577 

52 

0.90757 

67 

1.16937 

82 

1.43117 

8 

0.13963 

23 

0.40143 

38 

0.66323 

53 

0.92502 

68 

1.18682 

83 

1.44862 

9 

0 . 1570 S 

24 

0.41888 

39 

0.68068 

54 

0.94248 

69 

1.20428 

84 

1.46608 

10 

0.17453 

25 

0.43633 

40 

0.69813 

55 

0.95993 

70 

1.22173 

85 

1.48353 

11 

0.19199 

26 

0.45379 

41 

0.71558 

56 

0.97738 

71 

1.23918 

86 

1.50098 

12 

0.20944 

27 

0.47124 

42 

0.73304 

57 

0.99484 

72 

1.25664 

87 

1.51844 

13 

0.22689 

28 

0.48869 

43 

0.75049 

58 

1.01229 

73 

1.27409 

88 

1.53589 

14 

0.24435 

29 

0.50615 

44 

0.76794 

59 

1.02974 

74 

1.29154 

89 

1.55334 

15 

0.26180 

30 

0.52360 

45 

0.78540 

60 

1.04720 

75 

1.30900 

90 

1.57080 


1° = .01745329 rad. 
1' = .0002908882 rad. 
1" = .0000048481368 rad. 


log .01745329 = 8.24187737 - 10. 
log .0002908882 = 6.46372612 - 10. 
log .0000048481368 = 4.68557487 - 10. 


MINUTES IN RADIANS 


1 ' 

0.00029 

11 ' 

0.00320 

2 

0.00058 

12 

0.00349 

3 

0.00087 

13 

0.00378 

4 

0.00116 

14 

0.00407 

5 

0.00145 

15 

0.00436 

6 

0.00175 

16 

0.00465 

7 

0.00204 

17 

0.00495 

8 

0.00233 

18 

0.00524 

9 

0.00262 

19 

0.00553 

10 

0.00291 

20 

0.00582 


21 ' 

0.00611 

31 ' 

0.00902 

22 

0.00640 

32 

0.00931 

23 

0.00669 

33 

0.00960 

24 

0.00698 

34 

0.00989 

25 

0.00727 

35 

0.01018 

26 

0.00756 

36 

0.01047 

27 

0.00785 

37 

0.01076 

28 

0.00814 

38 

0.01105 

29 

0.00844 

39 

0.01134 

30 

0.00873 

40 

0.01164 


41 ' 

0.01193 

51 ' 

0.01484 

42 

0.01222 

52 

0.01513 

43 

0.01251 

53 

0.01542 

44 

0.01280 

54 

0.01571 

45 

0.01309 

55 

0.01600 

46 

0.01338 

56 

0.01629 

47 

0.01367 

57 

0.01658 

48 

0.01396 

58 

0.01687 

49 

0.01425 

59 

0.01716 

50 

0.01454 

60 

0.01745 


[ 121 ] 




















































XIV. NATURAL OR NAPERIAN LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 

0.0 0000 

0995 

1980 

2956 

3922 

4879 

5827 

6766 

7696 

8618 

1.1 

0.0 9531 

*0436 

*1333 

*2222 

*3103 

*3976 

*4842 

*5700 

*6551 

*7395 

1.2 

0.1 8232 

9062 

9885 

*0701 

*1511 

*2314 

*3111 

*3902 

*4686 

*5464 

1.3 

0.2 6236 

7003 

7763 

8518 

9267 

*0010 

*0748 

*1481 

*2208 

*2930 

1.4 

0.3 3647 

4359 

5066 

5767 

6464 

7156 

7844 

8526 

9204 

9878 

1.5 

0.4 0547 

1211 

1871 

2527 

3178 

3825 

4469 

5108 

5742 

6373 

1.6 

0.4 7000 

7623 

8243 

8858 

9470 

*0078 

*0682 

*1282 

*1879 

*2473 

1.7 

0.5 3063 

3649 

4232 

4812 

5389 

5962 

6531 

7098 

7661 

8222 

1.8 

0.5 8779 

9333 

9884 

*0432 

*0977 

*1519 

*2058 

*2594 

*3127 

*3658 

1.9 

0.6 4185 

4710 

5233 

5752 

6269 

6783 

7294 

7803 

8310 

8813 

2.0 

0.6 9315 

9813 

*0310 

*0804 

*1295 

*1784 

*2271 

*2755 

*3237 

*3716 

2.1 

0.7 4194 

4669 

5142 

5612 

6081 

6547 

7011 

7473 

7932 

8390 

2.2 

0.7 8846 

9299 

9751 

*0200 

*0648 

*1093 

*1536 

*1978 

*2418 

*2855 

2.3 

0.8 3291 

3725 

4157 

4587 

5015 

5442 

5866 

6289 

6710 

7129 

2.4 

0.8 7547 

7963 

8377 

8789 

9200 

9609 

*0016 

*0422 

*0826 

*1228 

2.5 

0.9 1629 

2028 

2426 

2822 

3216 

3609 

4001 

4391 

4779 

5166 

2.6 

5551 

5935 

6317 

6698 

7078 

7456 

7833 

8208 

8582 

8954 

2.7 

0.9 9325 

9695 

*0063 

*0430 

*0796 

*1160 

*1523 

*1885 

*2245 

*2604 

2.8 

1.0 2962 

3318 

3674 

4028 

4380 

4732 

5082 

5431 

5779 

6126 

2.9 

6471 

6815 

7158 

7500 

7841 

8181 

8519 

8856 

9192 

9527 

3.0 

1.0 9861 

*0194 

*0526 

*0856 

*1186 

*1514 

*1841 

*2168 

*2493 

*2817 

3.1 

1.1 3140 

3462 

3783 

4103 

4422 

4740 

5057 

5373 

5688 

6002 

3.2 

6315 

6627 

6938 

7248 

7557 

7865 

8173 

8479 

8784 

9089 

3.3 

1.1 9392 

9695 

9996 

*0297 

*0597 

*0896 

*1194 

*1491 

*1788 

*2083 

3.4 

1.2 2378 

2671 

2964 

3256 

3547 

3837 

4127 

4415 

4703 

4990 

3.5 

5276 

5562 

5846 

6130 

6413 

6695 

6976 

7257 

7536 

7815 

3.6 

1.2 8093 

8371 

8647 

8923 

9198 

9473 

9746 

*0019 

*0291 

*0563 

3.7 

1.3 0833 

1103 

1372 

1641 

1909 

2176 

2442 

2708 

2972 

3237 

3.8 

3500 

3763 

4025 

4286 

4547 

4807 

5067 

5325 

5584 

5841 

3.9 

6098 

6354 

6609 

6864 

7118 

7372 

7624 

7877 

8128 

8379 

4.0 

1.3 8629 

8879 

9128 

9377 

9624 

9872 

*0118 

*0364 

*0610 

*0854 

4.1 

1.4 1099 

1342 

1585 

1828 

2070 

2311 

2552 

2792 

3031 

3270 

4.2 

3508 

3746 

3984 

4220 

4456 

4692 

4927 

5161 

5395 

5629 

4.3 

5862 

6094 

6326 

6557 

6787 

7018 

7247 

7476 

7705 

7933 

4.4 

1.4 8160 

8387 

8614 

8840 

9065 

9290 

9515 

9739 

9962 

*0185 

4.5 

1.5 0408 

0630 

0851 

1072 

1293 

1513 

1732 

1951 

2170 

2388 

4.6 

2606 

2823 

3039 

3256 

3471 

3687 

3902 

4116 

4330 

4543 

4.7 

4756 

4969 

5181 

5393 

5604 

5814 

6025 

6235 

6444 

6653 

4.8 

6862 

7070 

7277 

7485 

7691 

7898 

8104 

8309 

8515 

8719 

4.9 

1.5 8924 

9127 

9331 

9534 

9737 

9939 

*0141 

*0342 

*0543 

*0744 

5.0 

1.6 0944 

1144 

1343 

1542 

1741 

1939 

2137 

2334 

2531 

2728 

5.1 

2924 

3120 

3315 

3511 

3705 

3900 

4094 

4287 

4481 

4673 

5.2 

4866 

5058 

5250 

5441 

5632 

5823 

6013 

6203 

6393 

6582 

5.3 

6771 

6959 

7147 

7335 

7523 

7710 

7896 

8083 

8269 

8455 

5.4 

1.6 8640 

8825 

9010 

9194 

9378 

9562 

9745 

9928 

*0111 

*0293 

5.5 

1.7 0475 

0656 

0838 

1019 

1199 

1380 

1560 

1740 

1919 

2098 

5.6 

2277 

2455 

2633 

2811 

2988 

3166 

3342 

3519 

3695 

3871 

5.7 

4047 

4222 

4397 

4572 

4746 

4920 

5094 

5267 

5440 

5613 

5.8 

5786 

5958 

6130 

6302 

6473 

6644 

6815 

6985 

7156 

7326 

5.9 

7495 

7665 

7834 

8002 

8171 

8339 

8507 

8675 

8842 

9009 

6.0 

1.7 9176 

9342 

9509 

9675 

9840 

*0006 

*0171 

*0336 

*0500 

*0665 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


[ 122 ] 






































































































XIV. NATURAL OR NAPERIAN LOGARITHMS 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6.0 

1.7 9176 

9342 

9509 

9675 

9840 

*0006 

*0171 

*0336 

*0500 

*0665 

6.1 

1.8 0829 

0993 

1156 

1319 

1482 

1645 

1808 

1970 

2132 

2294 

6.2 

2455 

2616 

2777 

2938 

3098 

3258 

3418 

3578 

3737 

3896 

6.3 

4055 

4214 

4372 

4530 

4688 

4845 

5003 

5160 

5317 

5473 

6.4 

5630 

5786 

5942 

6097 

6253 

6408 

6563 

6718 

6872 

7026 

6.5 

7180 

7334 

7487 

7641 

7794 

7947 

8099 

8251 

8403 

8555 

6.6 

1.8 8707 

8858 

9010 

9160 

9311 

9462 

9612 

9762 

9912 

*0061 

6.7 

1.9 0211 

0360 

0509 

0658 

0806 

0954 

1102 

1250 

1398 

1545 

6.8 

1692 

1839 

1986 

2132 

2279 

2425 

2571 

2716 

2862 

3007 

6.9 

3152 

3297 

3442 

3586 

3730 

3874 

4018 

4162 

4305 

4448 

7.0 

4591 

4734 

4876 

5019 

5161 

5303 

5445 

5586 

5727 

5869 

7.1 

6009 

6150 

6291 

6431 

6571 

6711 

6851 

6991 

7130 

7269 

7.2 

7408 

7547 

7685 

7824 

7962 

8100 

8238 

8376 

8513 

8650 

7.3 

1.9 8787 

8924 

9061 

9198 

9334 

9470 

9606 

9742 

9877 

*0013 

7.4 

2.0 0148 

0283 

0418 

0553 

0687 

0821 

0956 

1089 

1223 

1357 

7.5 

1490 

1624 

1757 

1890 

2022 

2155 

2287 

2419 

2551 

2683 

7.6 

2815 

2946 

3078 

3209 

3340 

3471 

3601 

3732 

3862 

3992 

7.7 

4122 

4252 

4381 

4511 

4640 

4769 

4898 

5027 

5156 

5284 

7.8 

5412 

5540 

5668 

5796 

5924 

6051 

6179 

6306 

6433 

6560 

7.9 

6686 

6813 

6939 

7065 

7191 

7317 

7443 

7568 

7694 

7819 

8.0 

7944 

8069 

8194 

8318 

8443 

8567 

8691 

8815 

8939 

9063 

8.1 

2.0 9186 

9310 

9433 

9556 

9679 

9802 

9924 

*0047 

*0169 

*0291 

8.2 

2.1 0413 

0535 

0657 

0779 

0900 

1021 

1142 

1263 

1384 

1505 

8.3 

1626 

1746 

1866 

1986 

2106 

2226 

2346 

2465 

2585 

2704 

8.4 

2823 

2942 

3061 

3180 

3298 

3417 

3535 

3653 

3771 

3889 

8.5 

4007 

4124 

4242 

4359 

4476 

4593 

4710 

4827 

4943 

5060 

8.6 

5176 

5292 

5409 

5524 

5640 

5756 

5871 

5987 

6102 

6217 

8.7 

6332 

6447 

6562 

6677 

6791 

6905 

7020 

7134 

7248 

7361 

8.8 

7475 

7589 

7702 

7816 

7929 

8042 

8155 

8267 

8380 

8493 

8.9 

8605 

8717 

8830 

8942 

9054 

9165 

9277 

9389 

9500 

9611 

9.0 

2.1 9722 

9834 

9944 

*0055 

*0166 

*0276 

*0387 • 

*0497 

*0607 

*0717 

9.1 

2.2 0827 

0937 

1047 

1157 

1266 

1375 

1485 

1594 

1703 

1812 

9.2 

1920 

2029 

2138 

2246 

2354 

2462 

2570 

2678 

2786 

2894 

9.3 

3001 

3109 

3216 

3324 

3431 

3538 

3645 

3751 

3858 

3965 

9.4 

4071 

4177 

4284 

4390 

4496 

4601 

4707 

4813 

4918 

5024 

9.5 

5129 

5234 

5339 

5444 

5549 

5654 

5759 

5863 

5968 

6072 

9.6 

6176 

6280 

6384 

6488 

6592 

6696 

6799 

6903 

7006 

7109 

9.7 

7213 

7316 

7419 

7521 

7624 

7727 

7829 

7932 

8034 

8136 

9.8 

8238 

8340 

8442 

8544 

8646 

8747 

8849 

8950 

9051 

9152 

9.9 

2.2 9253 

9354 

9455 

9556 

9657 

9757 

9858 

9958 

*0058 

*0158 

10.0 

2.3 0259 

0358 

0458 

0558 

0658 

0757 

0857 

0956 

1055 

1154 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


Note 1. The base for natural logarithms is e = 2.71828 18284 59045 • • • . 

log e 10 = 2.3025 8509. log 10 e = 0.4342 9448. (1) 

Note 2. If N > 10 or N < 1, then we may write N = P ■ 10 fc where k is an integer and 
1 P < 10. Then, to find log e N, use the following relation with log e P obtained from the 
preceding table and logelO obtained from (1): 

log e N = log e (P • 10*) = loggP + k log e 10. 


[ 123 ] 


























































































XV. IMPORTANT CONSTANTS AND THEIR LOGARITHMS 


Description 

Value 

logio 

7T 


3.14159265 


0.49714987 


e = Naperian Base 


2.71828183 


0.43429448 


M = logio e 


0.43429448 


9.63778431 - 

10 

1 + M = log e 10 


2.30258509 


0.36221569 


180 -r 7 r = degrees in 1 radian 


57.2957795 


1.75812263 


7 r -T- 180 = radians in 1° 


0.01745329 


8.24187737 - 

10 

7 r -v- 10800 = radians in 1' 


0.0002908882 


6.46372612 - 

10 

t r -T- 648000 = radians in 1" 


0.000004848136811095 

4.68557487 - 

10 

Values with Extreme Accuracy 

7T = 3.14159 

26535 89793 23846 

26433 

83280 


logio 7r = 0.49714 

98726 94133 85435 

12682 

88291 


loge 7T = 1.14472 

98858 49400 17414 

34273 

51353 


e = 2.71828 

18284 59045 23536 

02874 

71353 


M = 0.43429 

44819 03251 82765 

11289 

18917 


1 -j- M = 2.30258 

50929 94045 68401 

79914 

54684 


logio M = 9.63778 

43113 00536 78912 

- 10 




XVI. LOGARITHMS FOR COMPUTING COMPOUND INTEREST* 


i% 

logio (1 + 0 

i% 

logio (4 + 0 

0 + 1/12 

.00036 

17613 

55281 

2 + 1/4 

.00966 

33166 

79379 

1/8 

.00054 

25290 

92294 

3/8 

.01019 

39147 

68475 

1/6 

.00072 

32216 

19096 

1/2 

.01072 

38653 

91773 

5/24 

.00090 

38389 

98245 

5/8 

.01125 

31701 

27497 

1/4 

.00108 

43812 

92220 

3/4 

.01178 

18305 

48107 

0 + 7/24 

.00126 

48485 

63424 

2 + 7/8 

.01230 

98482 

20326 

1/3 

.00144 

52408 

74181 

3 + 0 

.01283 

72247 

05172 

3/8 

.00162 

55582 

86737 

1/4 

.01389 

00603 

28439 

5/12 

.00180 

58008 

63262 

1/2 

.01494 

03497 

92937 

1/2 

.00216 

60617 

56508 

3/4 

.01598 

81053 

84130 

0 + 7/12 

.00252 

60240 

49724 

4+0 

.01703 

33392 

98780 

5/8 

.00270 

58933 

75925 

1/4 

.01807 

60636 

45795 

2/3 

.00288 

56882 

37488 

1/2 

.01911 

62904 

47073 

3/4 

.00324 

50548 

13147 

3/4 

.02015 

40316 

38333 

7/8 

.00378 

35477 

30127 

5 + 0 

.02118 

92990 

69938 

1 + 0 

.00432 

13737 

82643 

5+1/4 

.02222 

21045 

07706 

1/8 

.00485 

85346 

20329 

1/2 

.02325 

24596 

33711 

1/4 

.00539 

50318 

86706 

3/4 

.02428 

03760 

470S0 

3/8 

.00593 

08672 

19212 

6+0 

.02530 

58652 

64770 

1/2 

.00646 

60422 

49232 

1/4 

.02632 

89387 

22349 

1 + 5/8 

.00700 

05586 

02125 

6+1/2 

.02734 

96077 

74757 

3/4 

.00753 

44178 

97258 

7 + 0 

.02938 

37776 

85210 

7/8 

.00806 

76217 

48033 

1/2 

.03140 

84642 

51624 

2 + 0 

.00860 

01717 

61918 

8 + 0 

.03342 

37554 

86950 

1/8 

.00913 

20695 

40^72 

1/2 

.03542 

97381 

84548 


* For a more complete table see Glover’s Interest and Logarithmic Tables ; George Wahr, 
publisher. 


[ 124 ] 












































. 








4 































